Advances in Matrices, Finite And

Home , Carlos Conca, Numerical linear algebra

Journal of Applied Mathematics

Advances in Matrices, Finite and

Guest Editors: P. N. Shivakumar, Panayiotis Psarrakos, K. C. Sivakuma r, Yang Zhang, and Carlos da Fonseca Advances in Matrices, Finite and Infinite, with Applications 2014 JournalofAppliedMathematics

Advances in Matrices, Finite and Infinite, with Applications 2014

This is a special issue published in “Journal of Applied Mathematics.” All articles are open access articles distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Editorial Board

Saeid Abbasbandy, Iran Ru-Dong Chen, China Laura Gardini, Italy Mina B. Abd-El-Malek, Egypt Zhang Chen, China Bernard J. Geurts, The Netherlands Mohamed A. Abdou, Egypt Zhi-Zhong Chen, Japan Sandip Ghosal, USA Subhas Abel, India Xinkai Chen, Japan Pablo Gonzlez-Vera, Spain Jnos Abonyi, Hungary Rushan Chen, China Alexander N. Gorban, UK M. Montaz Ali, South Africa Ke Chen, UK Laurent Gosse, Italy Mohammad R, Aliha, Iran Eric Cheng, Hong Kong Keshlan S. Govinder, South Africa Carlos J. S. Alves, Portugal Ching-Hsue Cheng, Taiwan Said R. Grace, Egypt Mohamad Alwash, USA Qi Cheng, USA Jose L. Gracia, Spain Igor Andrianov, Germany Chin-Hsiang Cheng, Taiwan Maurizio Grasselli, Italy Boris Andrievsky, Russia Jin Cheng, China Zhi-Hong Guan, China Whye-Teong Ang, Singapore Hui Cheng, China Nicola Guglielmi, Italy Abul-Fazal M. Arif, Saudi Arabia Francisco Chiclana, UK Fred´ eric´ Guichard, Canada Sabri Arik, Turkey Jen-Tzung Chien, Taiwan Kerim Guney, Turkey Ali R. Ashrafi, Iran Han H. Choi, Republic of Korea Shu-Xiang Guo, China Allaberen Ashyralyev, Turkey Sazzad H. Chowdhury, Malaysia Vijay Gupta, India FrancisT.K.Au,HongKong Hung-Yuan Chung, Taiwan SamanK.Halgamuge,Australia Francesco Aymerich, Italy Angelo Ciaramella, Italy Abdelmagid S. Hamouda, Qatar Seungik Baek, USA Pedro J. Coelho, Portugal Bo Han, China Olivier Bahn, Canada Carlos Conca, Chile Maoan Han, China Antonio Bandera, Spain Vitor Costa, Portugal Pierre Hansen, Canada Jean-Pierre Barbot, France Livija Cveticanin, Serbia Ferenc Hartung, Hungary Mostafa Barigou, UK Binxiang Dai, China Jesper H. Hattel, Denmark Roberto Barrio, Spain Youjun Deng , China Xiao-Qiao He, China Alfredo Bellen, Italy Orazio Descalzi, Chile Yuqing He, China Jafar Biazar, Iran Raffaele Di Gregorio, Italy Onesimo Hernandez-Lerma, Mexico Anjan Biswas, Saudi Arabia Kai Diethelm, Germany Luis J. Herrera, Spain Abdellah Bnouhachem, Morocco Daniele Dini, UK J. C. D. Hoenderkamp, The Netherlands Gabriele Bonanno, Italy Urmila Diwekar, USA Thomas H¨hne, Germany Stephane P.A. Bordas, USA Vit Dolejsi, Czech Republic Wei-Chiang Hong, Taiwan James Robert Buchanan, USA Bo-Qing Dong, China Sun-Yuan Hsieh, Taiwan Humberto Bustince, Spain Rafael Escarela-Perez, Mexico Ying Hu, France Alberto Cabada, Spain Magdy A. Ezzat, Egypt Ning Hu, Japan Xiao Chuan Cai, USA Meng Fan, China Jianguo Huang, China Piermarco Cannarsa, Italy Ya Ping Fang, China Dan Huang, China Jinde Cao, China Didier Felbacq, France Zhilong L. Huang, China Yijia Cao, China AntonioJ.M.Ferreira,Portugal Ting-Zhu Huang, China Zhenfu Cao, China Michel Fliess, France Zhenkun Huang, China Alexandre Carvalho, Brazil Marco A. Fontelos, Spain Mustafa Inc, Turkey Song Cen, China Dimitris Fotakis, Greece Gerardo Iovane, Italy Shih-sen Chang, China Tomonari Furukawa, USA Anuar Ishak, Malaysia Tai-Ping Chang, Taiwan Maria Gandarias, Spain Takeshi Iwamoto, Japan Wei-Der Chang, Taiwan Xiao-wei Gao, China Lucas J. Sanchez,´ Spain Shuenn-Yih Chang, Taiwan Huijun Gao, China George Jaiani, Georgia Kripasindhu Chaudhuri, India Xin-Lin Gao, USA GunHee Jang, Republic of Korea Zhongxiao Jia, China Dongfang Liang, UK Roberto Natalini, Italy Daqing Jiang, China Ching-Jong Liao, Taiwan Srinivasan Natesan, India Haijun Jiang, China Yong-Cheng Lin, China Tatsushi Nishi, Japan Jianjun Jiao, China Chong Lin, China Andreas Ochsner,¨ Australia Xing Jin, China Chein-Shan Liu, Taiwan Wlodzimierz Ogryczak, Poland Zhen Jin, China Kang Liu, USA Roger Ohayon, France Zlatko Jovanoski, Australia Zhijun Liu, China Javier Oliver, Spain Tadeusz Kaczorek, Poland Yansheng Liu, China Soontorn Oraintara, USA Ido Kanter, Israel Peter Liu, Taiwan Donal O’Regan, Ireland Abdul Hamid Kara, South Africa Weiqing Liu, China Martin Ostoja-Starzewski, USA Hamid Reza Karimi, Norway Tao Liu, China Turgut Ozi,¨ Turkey Dimitrios A. Karras, Greece Shutian Liu, China Claudio Padra, Argentina Ihsan Kaya, Turkey Fei Liu, China Vincent Pagneux, France Dogan Kaya, Turkey Chongxin Liu, China Reinaldo Martinez Palhares, Brazil Chaudry M. Khalique, South Africa Zhengrong Liu, China Quan K. Pan, China Khalil Khanafer, USA Zhuangjian Liu, Singapore Endre Pap, Serbia Adem Kılıc¸man, Malaysia Jose L. Lopez,´ Spain Manuel Pastor, Spain Hyunsung Kim, Republic of Korea Shiping Lu, China Giuseppe Pellicane, South Africa Younjea Kim, Republic of Korea Hongbing Lu, China Francesco Pellicano, Italy Jong Hae Kim, Republic of Korea Benzhuo Lu, China Juan Manuel Pena,˜ Spain Kazutake Komori, Japan Henry Horng-Shing Lu, Taiwan Jian-Wen Peng, China Vassilis Kostopoulos, Greece Yuan Lu, China Ricardo Perera, Spain Jisheng Kou, China Changfeng Ma, China Fernando Perez-Cruz, Spain Roberto A. Kraenkel, Brazil Ruyun Ma, China Malgorzata Peszynska, USA Vadim A. Krysko, Russia Li Ma, China Allan C. Peterson, USA Jin L. Kuang, Singapore Lifeng Ma, China Vu Ngoc Phat, Vietnam V. Ku m a r a n , In d i a Nazim I. Mahmudov, Turkey Andrew Pickering, Spain Miroslaw Lachowicz, Poland Oluwole D. Makinde, South Africa Hector Pomares, Spain Hak-Keung Lam, UK Francisco J. Marcelln, Spain Mario Primicerio, Italy Tak-Wah Lam, Hong Kong Giuseppe Marino, Italy Morteza Rafei, The Netherlands Heung-Fai Lam, Hong Kong Guiomar Mart´ın-Herrn, Spain Laura Rebollo-Neira, UK Luciano Lamberti, Italy Carlos Mart´ın-Vide, Spain Roberto Reno,` Italy PeterGLLeach,Cyprus Alessandro Marzani, Italy Juan A. Rodriguez-Velazquez,´ Spain Jaehong Lee, Republic of Korea Nikos E. Mastorakis, Bulgaria Ignacio Rojas, Spain Usik Lee, Republic of Korea Nicola Mastronardi, Italy Carla Roque, Portugal Myung-Gyu Lee, Republic of Korea Panayotis T. Mathiopouloss, Greece Debasish Roy, India Wen-Chuan Lee, Taiwan Gianluca Mazzini, Italy Imre J. Rudas, Hungary Jinsong Leng, China Marta Mazzocco, UK Abbas Saadatmandi, Iran Xiang Li, China Michael McAleer, The Netherlands Kunihiko Sadakane, Japan Yongkun Li, China Stephane Metens, France Samir Saker, Egypt Wan-Tong Li, China Michael Meylan, Australia Juan JoseSalazarGonz´ alez,´ Spain Qingdu Li, China Fan Min, China Miguel A. F. Sanjuan, Spain Hua Li, Singapore Alain Miranville, France Bogdan Sasu, Romania Lixiang Li, China Ram N. Mohapatra, USA Richard Saurel, France Wenlong Li, Hong Kong Gisele Mophou, France Wolfgang Schmidt, Germany Shuai Li, Hong Kong Cristinel Mortici, Romania Jose R. Serrano, Spain Yan Liang , China Jaime E. Munoz Rivera, Brazil Mehmet Sezer, Turkey Jin Liang, China Javier Murillo, Spain Naseer Shahzad, Saudi Arabia Pengjian Shang, China Heng Wang, Singapore Bin Yang, China Hui-Shen Shen, China Yaonan Wang , China Guowei Yang, China Jian Hua Shen, China Qing-Wen Wang, China Chunyu Yang, China Yong Shi, China Guangchen Wang, China Her-Terng Yau, Taiwan Yasuhide Shindo, Japan Baolin Wang, China Wei-Chang Yeh, Taiwan Patrick Siarry, France Mingxin Wang, China Guan H. Yeoh, Australia Fernando Simoes,˜ Portugal Dongqing Wang, China Chih-Wei Yi, Taiwan TheodoreE.Simos,Greece Youqing Wang, China Simin Yu, China Francesco Soldovieri, Italy Xiang Wang, China Bo Yu, China Abdel-Maksoud A. Soliman, Egypt Shuming Wang, Singapore Xiaohui Yuan, China Qiankun Song, China Yuh-Rau Wang , Taiwan Jinyun Yuan, Brazil Xinyu Song, China Pei-Guang Wang, China Rafal Zdunek, Poland Yuri N. Sotskov, Belarus Junzo Watada, Japan Ashraf Zenkour, Egypt Niclas Str¨mberg, Sweden Jinjia Wei, China Guisheng Zhai, Japan Ray K.L. Su, Hong Kong Guoliang Wei, China Jianming Zhan, China Housheng Su, China Junjie Wei, China Meng Zhan, China Jitao Sun, China Li Weili, China Long Zhang, China Wenyu Sun, China Martin Weiser, Germany Ke Zhang, China Chengjun Sun, Hong Kong Frank Werner, Germany Sheng Zhang, China Toshio Tagawa, Japan Man-Leung Wong, Hong Kong Jifeng Zhang, China Ying Tan, China Min-Hsien Wu, Taiwan Heping Zhang, China San-Yi Tang, China Wei Wu, China Henggui Zhang, UK XianHua Tang, China Shi-Liang Wu, China Liang Zhang, China Zhidong Teng, China Cheng Wu, China Jian-gang Zhang, China Engang Tian, China Shanhe Wu, China Jingxin Zhang, Australia Alexander Timokha, Norway Yonghui X ia, China Zhihua Zhang, China Yiying Tong, USA Tiecheng Xia, China Shan Zhao, USA Hossein Torkaman, Iran Gongnan Xie, China Chongbin Zhao, Australia Mariano Torrisi, Italy Xuejun Xie, China Renat Zhdanov, USA Jung-Fa Tsai, Taiwan Wei Xu, China Dong Zheng, USA Ch. Tsitouras, Greece Daoyi Xu, China Huaichun Zhou, China Antonia Tulino, USA Zhiqiang Xu, China Bin Zhou, China Sergey Utyuzhnikov, UK Yuesheng Xu, USA William Zhu, China Kuppalapalle Vajravelu, USA Fuzhen Xuan, China Xinqun Zhu, Australia Alvaro Valencia, Chile Gregory S. Yablonsky, USA Quanxin Zhu, China Erik Van Vleck, USA Chao Yan, USA Goangseup Zi, Republic of Korea Ezio Venturino, Italy Chao Yang, China Zhiqiang Zuo, China Jesus Vigo-Aguiar, Spain Suh-Yuh Yang, Taiwan Michael N. Vrahatis, Greece Chao-Tung Yang, Taiwan Contents

Advances in Matrices, Finite and Infinite, with Applications 2014,P.N.Shivakumar, Panayiotis J. Psarrakos, K. C. Sivakumar, Yang Zhang, and Carlos M. da Fonseca Volume 2014, Article ID 518487, 1 page

On Comparison Theorems for Splittings of Different Semimonotone Matrices, Shu-Xin Miao and Yang Cao Volume 2014, Article ID 329490, 4 pages

Some Results on Characterizations of Matrix Partial Orderings,HongxingWangandJinXu Volume 2014, Article ID 408457, 6 pages

A Test Matrix for an Inverse Eigenvalue Problem,G.M.L.Gladwell,T.H.Jones,andN.B.Willms Volume2014,ArticleID515082,6pages

Natural Filtrations of Infinite-Dimensional Modular Contact Superalgebras,QiangMu Volume 2014, Article ID 601847, 9 pages

Nonnegative Combined Matrices,RafaelBru,MariaT.Gasso,´ Isabel Gimenez,´ and Maximo´ Santana Volume 2014, Article ID 182354, 5 pages Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2014, Article ID 518487, 1 page http://dx.doi.org/10.1155/2014/518487

Editorial Advances in Matrices, Finite and Infinite, with Applications 2014

P. N. Shivakumar,1 Panayiotis J. Psarrakos,2 K. C. Sivakumar,3 Yang Zhang,1 and Carlos M. da Fonseca4

1 Department of Mathematics, University of Manitoba, Winnipeg, MB, Canada R3T 2N2 2 Department of Mathematics, School of Applied Mathematical & Physical Sciences, National Technical University of Athens, Zografou Campus, 15780 Athens, Greece 3 Department of Mathematics, Indian Institute of Technology Madras, Chennai 600 036, India 4 Department of Mathematics, Kuwait University, 13060 Safat, Kuwait

Correspondence should be addressed to P. N. Shivakumar; [email protected]

Received 19 August 2014; Accepted 19 August 2014; Published 7 September 2014

Copyright © 2014 P. N. Shivakumar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Matrix theory either finite or infinite has increasingly proved a simple derivation of a set of explicit expressions is provided to be a key element in many different modern scientific for the components of a Jacobi matrix of order 𝑛×𝑛.This fields, far beyond its natural mathematical environment. This matrix has the property that its eigenvalues come from the volume exposes the growing sophistication of the techniques set {0,2,...,2𝑛 − 2} while also satisfying the additional involving matrices as well as some of many applications. condition that the eigenvalues of the leading principal (𝑛 − Classification of modular Lie algebras has recently been 1) × (𝑛 − 1) submatrix belong to the set {1,3,...,2𝑛−3}.As the subject of many authors. In particular, alteration tech- an application, an explicit solution for a spring-mass inverse niques play a major role. Natural filtration for Cartan types problem is presented. has been shown to be invariant in infinite dimensional The combined matrix of a nonsingular matrix 𝐴 is defined −1 𝑇 case. In this issue, Q. Mu in “Natural filtrations of infinite- as 𝐶(𝐴) = 𝐴 ∘(𝐴 ) ,where∘ means the Hadamard product. dimensional modular contact superalgebras”establishesthat Combined matrices appear in the chemical literature, where the natural filtration is invariant under automorphisms in the they represent the relative gain array. Furthermore, the case of Lie algebras. The author uses ad-nilpotent elements study of combined matrices yields the relation between the techniques. eigenvalues and diagonal entries of a diagonalizable matrix. It is well known that rank of matrix plays an important It is well known that the row and column sums of a combined role in matrix theory and has many applications in other matrix are always equal to one, and, consequently, if 𝐶(𝐴) areas. In the paper “Some results on characterizations of is entrywise nonnegative, then it has interesting properties matrix partial orderings”byH.WangandJ.Xu,fivematrix and applications since it is a doubly stochastic matrix. In C𝑚×𝑛 partial orderings defined in are considered and the the paper “Nonnegative combined matrices”ofR.Bruetal., (left/right- and both sided-) star and sharp partial orderings the matrices which have nonnegative combined matrices are are characterized by applying rank equalities. investigated. In particular, combined matrices of different Comparison theorems between the spectral radii of classes of matrices, such as totally positive and totally negative different matrices are useful tools for judging the efficiency matrices, and also when 𝐴 is totally nonnegative and totally of preconditioned. In S.-X. Miao and Y. Cao’s paper “On nonpositive, are studied. comparison theorems for splittings of different semimonotone matrices,” the authors gave some comparisons for the spectral P. N . S hiv akumar radii of matrices arising from proper splittings of different Panayiotis J. Psarrakos semimonotone matrices. K. C. Sivakumar In the work reported by G. M. L Gladwell et al. in Yang Zhang the paper “A test matrix for an inverse eigenvalue problem,” Carlos M. da Fonseca Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2014, Article ID 329490, 4 pages http://dx.doi.org/10.1155/2014/329490

Research Article On Comparison Theorems for Splittings of Different Semimonotone Matrices

Shu-Xin Miao1 and Yang Cao2

1 College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China 2 School of Transportation, Nantong University, Nantong 226019, China

Correspondence should be addressed to Shu-Xin Miao; [email protected]

Received 26 March 2014; Revised 24 June 2014; Accepted 24 June 2014; Published 6 July 2014

Academic Editor: Yang Zhang

Copyright © 2014 S.-X. Miao and Y. Cao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Comparison theorems between the spectral radii of different matrices are useful tools for judging the efficiency of preconditioners. In this paper, some comparison theorems for the spectral radii of matrices arising from proper splittings of different semimonotone matrices are presented.

1. Introduction and Preliminaries 𝑁(𝐴) aretherangeandkernelof𝐴,respectively.Let𝜌(𝐶) be the spectral radius of the real square matrix 𝐶; then, for the 𝑂 Let be the null matrix with suitable size. The notation proper splitting 𝐴=𝑈−𝑉, the iterative scheme (2)converges 𝐴≥𝑂(𝐴>𝑂)denotes that all entries of matrix 𝐴 are † to the minimal norm least square solution 𝑥=𝐴𝑏 of (1)for nonnegative (positive), and in this case matrix 𝐴 is called 0 † 𝑚×𝑛 𝐴, 𝐵 any initial vector 𝑥 if and only if 𝜌(𝑈 𝑉) < 1 [4,Corollary nonnegative (positive). For two real matrices, , 𝐴=𝑈−𝑉 𝐴≥𝐵(𝐴>𝐵)means that 𝐴−𝐵≥𝑂 (𝐴−𝐵>𝑂).Thesame 1]. In this case, we say that the proper splitting is 𝑚×𝑛 a convergent splitting. Moreover, the fact that 𝑈=𝐴+𝑉is notation is valid for vectors. A real rectangular matrix 𝐴=𝑈−𝑉 𝐴 𝐴† ≥𝑂 𝐴† apropersplitting,as is a proper splitting, implies is said to be semimonotone if [1]; here is the 𝜌(𝐴†𝑉) < 1 𝐼+𝐴†𝑉 𝑈† = Moore-Penrose inverse of 𝐴,thatis,theuniquematrixwhich that and is invertible, so we have † † † (𝐼+𝐴†𝑉)−1𝐴† 𝑈†𝑉=(𝐼+𝐴†𝑉)−1𝐴†𝑉 satisfies the Moore-Penrose equation 𝐴𝐴 𝐴=𝐴, 𝐴 𝐴𝐴 = [1,Theorem2.2]and . 𝐴† (𝐴𝐴†)𝑇 =𝐴𝐴† (𝐴†𝐴)𝑇 =𝐴†𝐴 𝐵𝑇 The next lemma shows the relation between the eigenvalues , ,and ( denotes the † † transpose of 𝐵); see [2, 3]. of 𝑈 𝑉 and 𝐴 𝑉. Real rectangular linear system of the form Lemma 1 (see [1, Lemma 2.6]). Let 𝐴=𝑈−𝑉be a proper 𝐴𝑥=𝑏, (1) splitting of real 𝑚×𝑛matrix 𝐴.Let𝜇𝑖,1 ≤ 𝑖 ≤,and 𝑠 𝜆𝑗,1 ≤ † † where 𝐴 is a real 𝑚×𝑛matrix and 𝑏 is a real 𝑚-vector, appears 𝑗≤𝑠,betheeigenvaluesof𝑈 𝑉 and 𝐴 𝑉,respectively.Then in many areas of mathematics. For example, finite difference for every 𝑗,wehave1+𝜆𝑗 =0̸.Also,forevery𝑖, there exists 𝑗 discretization of partial differential equations with Neumann such that 𝜇𝑖 =(𝜆𝑗/(1 + 𝜆𝑗)) and, for every 𝑗, there exists 𝑖 such boundary conditions. Iterative methods for solving (1)canbe that 𝜆𝑗 =(𝜇𝑖/(1 − 𝜇𝑖)). formulated by the decomposition of 𝐴 as 𝐴=𝑈−𝑉[4], and the approximation solution of (1) is generated by For nonnegative matrix, there is a well-known result (𝑘+1) † (𝑘) † which is shown next. 𝑥 =𝑈𝑉𝑥 +𝑈 𝑏. (2) The decomposition 𝐴=𝑈−𝑉is called a proper splitting Lemma 2 (see [5,Theorem2.21]). Let 𝐴, 𝐵 be 𝑛×𝑛matrices; if 𝑅(𝐴) = 𝑅(𝑈) and 𝑁(𝐴) = 𝑁(𝑈) [4], where 𝑅(𝐴) and if 𝐴≥𝐵≥𝑂,then𝜌(𝐴) ≥ 𝜌(𝐵). 2 Journal of Applied Mathematics

Using the notation of nonnegative matrix, the proper Remark 5. Jena et al. [8] concluded that, for a proper weak regular and proper weak regular splittings, which are the regular splitting of real 𝑚×𝑛matrix 𝐴,theconvergence † naturalextensionsoftheregularandweakregularsplittings conditions are 𝐴 ≥𝑂and 𝑉≥𝑂,sotheconditionof of a real square matrix [5, 6], are defined as follows. Theorem 4 is weaker than that in [8]. To see this, let 2−10 200 010 Definition 3. For a real 𝑚×𝑛matrix 𝐴,thesplitting𝐴=𝑈−𝑉 𝐴=[ ], 𝑈[ ], 𝑉=[ ]. is called −3 2 0 −3 1 0 0−10 (4) (1)properregularsplittingifitisapropersplittingsuch † Then 𝐴 is a semimonotone matrix and 𝐴=𝑈−𝑉is a proper that 𝑈 ≥𝑂and 𝑉≥𝑂[7,Definition1],[8, † weak regular splitting of 𝐴.Itiseasytoseethat𝐴 𝑉≥𝑂and Definition 1.2]; † 𝜌(𝑈 𝑉) = 0.5,but <1 𝑉≥𝑂does not hold. (2)properweakregularsplittingoffirsttypeifitisa 𝑈† ≥𝑂 𝑈†𝑉≥𝑂 proper splitting such that and ; Let 𝐴1 and 𝐴2 be two semimonotone matrices and let proper weak regular splitting of second type if it is a 𝐴1 =𝑈1 −𝑉1 and 𝐴2 =𝑈2 −𝑉2 be proper splittings † † proper splitting such that 𝑈 ≥𝑂and 𝑉𝑈 ≥𝑂[7, of 𝐴1 and 𝐴2, respectively. In what follows, we will present † † Definition 1],8 [ ,Definition1.2]. the comparison results between 𝜌(𝑈1 𝑉1) and 𝜌(𝑈2 𝑉2).The comparison theorems for proper regular splittings are given It should be remarked that Jena et al. [8] only considered first. proper weak regular splitting of first type; they name it as proper weak regular splitting. The existence of the proper Theorem 6. Let 𝐴1 and 𝐴2 be two semimonotone matrices splitting is discussed in [4]; there is an example in [4]toshow and let 𝐴1 =𝑈1 −𝑉1 and 𝐴2 =𝑈2 −𝑉2 be proper regular † † how to construct such splitting. splittings of 𝐴1 and 𝐴2,respectively.If𝐴 ≥𝐴 and 𝑉2 ≥𝑉1, 𝐴=𝑈−𝑉 𝐴 2 1 Let be a proper regular splitting of ;Berman then 𝜌(𝑈†𝑉) < 1 and Plemmons in [4]showedthat if and only if † † † 𝜌(𝑈 𝑉 )≤𝜌(𝑈𝑉 )<1. 𝐴 ≥𝑂. Other convergence results of proper regular and/or 1 1 2 2 (5) weak regular splitting can be found in [8, 9]. Comparison Proof. As 𝐴1 and 𝐴2 are semimonotone matrices and 𝐴1 = theorems between the spectral radii of matrices are useful 𝑈1 −𝑉1 and 𝐴2 =𝑈2 −𝑉2 are proper regular splittings, it tools for analyzing the rate of convergence of iterative meth- † follows from [4]that𝜌(𝑈𝑖 𝑉𝑖)<1for 𝑖=1,2.Thusallwe ods or for judging the efficiency of preconditioners [8, 10–12]. 𝜌(𝑈†𝑉 )≤𝜌(𝑈†𝑉 ) There is also a connection to population dynamics [11]. Some need to show is 1 1 2 2 . 𝑖=1,2 𝐴†𝑉 comparison theorems of proper splittings of a semimonotone For ,notethatthematrices 𝑖 𝑖 are nonnegative; matrix are established recently in [8, 13]. Perron-Frobenius theorem (cf. [5]) states that the spectral 𝜌(𝐴†𝑉) 𝐴†𝑉 Ourbasicpurposehereistogiveanewconvergence radius 𝑖 𝑖 of 𝑖 𝑖 is an eigenvalue corresponding to a † theorem for proper weak regular splitting of a semimono- nonnegative eigenvector; then from Lemma 1, (𝜌(𝐴𝑖 𝑉𝑖)/(1 + † † † tone matrix and to derive the comparison theorems of 𝜌(𝐴𝑖 𝑉𝑖))) ≥ 0 is an eigenvalue of 𝑈𝑖 𝑉𝑖;hence,𝜌(𝑈𝑖 𝑉𝑖)≥ † † proper regular and proper weak regular splittings of different (𝜌(𝐴𝑖 𝑉𝑖 )/(1 + 𝜌(𝐴𝑖 𝑉𝑖))). Again, by Perron-Frobenius the- semimonotone matrices. The condition of new convergence † orem, 𝑈𝑖 𝑉𝑖 ≥𝑂implies existence of a nonnegative vector theorem is weaker than that in [8], and the comparison † † 𝑥(𝑥=0)̸ such that 𝑈 𝑉𝑖𝑥=𝜌(𝑈𝑉𝑖)𝑥.Then results generalized the corresponding results in [5, 8, 11]. 𝑖 𝑖 † The comparison results can be further used for judging the −1 𝜌(𝑈 𝑉) 𝐴†𝑉𝑥=(𝐼−𝑈†𝑉) 𝑈†𝑉𝑥= 𝑖 𝑖 𝑥 efficiency of the preconditioners. 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 † (6) 1−𝜌(𝑈𝑖 𝑉𝑖) 2. Main Results † † † implies (𝜌(𝑈𝑖 𝑉𝑖)/(1 − 𝜌(𝑈𝑖 𝑉𝑖))) ≥ 0 is an eigenvalue of 𝐴𝑖 𝑉𝑖; 𝜌(𝐴†𝑉) ≥ (𝜌(𝑈†𝑉)/(1 − 𝜌(𝑈†𝑉))) 𝜌(𝑈†𝑉)≤ Recall that the proper regular splitting of a semimonotone hence, 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 ;thatis, 𝑖 𝑖 (𝜌(𝐴†𝑉)/(1 + 𝜌(𝐴†𝑉))) matrix is a convergent splitting [4, 8].Forproperweakregular 𝑖 𝑖 𝑖 𝑖 . Therefore, we have splitting of a semimonotone matrix, we have the following 𝜌(𝐴†𝑉) convergence theorem. 𝜌(𝑈†𝑉)= 𝑖 𝑖 . 𝑖 𝑖 † (7) 1+𝜌(𝐴𝑖 𝑉𝑖) Theorem 4. Let 𝐴=𝑈−𝑉be a proper weak regular splitting 𝑚×𝑛 𝐴 𝐴† ≥𝑂 𝐴†𝑉≥𝑂 † † (ofanytype)ofreal matrix .If and , Note that 𝑉1 ≥𝑂;then,𝐴2 ≥𝐴1 and 𝑉2 ≥𝑉1 lead to † † † † then 𝐴2𝑉2 ≥𝐴1𝑉1 ≥𝑂,andLemma 2 yields 𝜌(𝐴1𝑉1)≤𝜌(𝐴2𝑉2). 𝑓(𝜆) = (𝜆/(1 +𝜆)) 𝑓(𝜆) 𝜌(𝐴†𝑉) Let ;then, is a strictly increasing † † † † 𝜌(𝑉𝑈 )=𝜌(𝑈𝑉) = <1. function for 𝜆>0.Hencetheinequality𝜌(𝑈1 𝑉1)≤𝜌(𝑈2 𝑉2) † (3) 1+𝜌(𝐴 𝑉) holds.

† † † Proof. Note that 𝐴 𝑉≥𝑂; the proof is essentially analogous Remark 7. The assumptions 𝐴2 ≥𝐴1 and 𝑉2 ≥𝑉1 † † to that in [8]. We omit it here. of Theorem 4 canbeweakenedas𝐴2𝑉2 ≥𝐴1𝑉1. Journal of Applied Mathematics 3

† † † † † † † † For different proper regular splittings of one semimono- 𝑈𝑖 𝐴𝑖𝐴𝑖 =𝑈𝑖 ,for𝑖=1,2,and𝐴1𝐴1𝑈1 =𝑈1 and 𝐴1𝑈1𝑈1 = 𝐴 † † † † † tone matrix , the following corollary is obtained. 𝐴1 (see, e.g., [3, Exercise 1.3(2)]). Using 𝑈1 −𝑈2 ≥𝐴1 −𝐴2 we obtain Corollary 8 (see [8, Theorem 3.2]). Let 𝐴 be a semimonotone 𝐴=𝑈 −𝑉 =𝑈 −𝑉 † † † † † † matrix and let 1 1 2 2 be two proper regular 𝑈2 𝑉2𝐴2 =𝑈2 (𝑈2 −𝐴2)𝐴2 =𝐴2 −𝑈2 splittings of 𝐴.If𝑉2 ≥𝑉1,then ≥𝐴† −𝑈† =𝑈† (𝑈 −𝐴 )𝐴† † † 1 1 1 1 1 1 𝜌(𝑈1 𝑉1)≤𝜌(𝑈2 𝑉2)<1. (8) (14) † † =𝑈1 𝑉1𝐴1 When we consider the monotone matrices, we have the † † following corollaries directly. =𝐴1𝑉1𝑈1 ≥𝑂.

Corollary 9 𝐴 𝐴 † † (see [11,Theorem4.2]). Let 1 and 2 be two For 𝑈 𝑉1 ≥𝑂and 𝑈 𝑉2 ≥𝑂, by Perron-Frobenius theorem 𝐴 =𝑈 −𝑉 𝐴 =𝑈 −𝑉 1 2 monotone matrices and let 1 1 1 and 2 2 2 be (cf. [5]), there exist two nonzero vectors 𝑥≥0and 𝑦≥0such 𝐴 𝐴 𝐴−1 ≥𝐴−1 regular splittings of 1 and 2,respectively.If 2 1 and that 𝑉2 ≥𝑉1,then † † 𝑇 † 𝑇 † −1 −1 𝑉1𝑈1 𝑥=𝜌(𝑈1 𝑉1)𝑥, 𝑦 𝑈2 𝑉2 =𝑦 𝜌(𝑈2 𝑉2). (15) 𝜌(𝑈1 𝑉1)≤𝜌(𝑈2 𝑉2)<1. (9) Thus Corollary 10. Let 𝐴1 and 𝐴2 be two monotone matrices and let 𝐴1 =𝑈1 −𝑉1 and 𝐴2 =𝑈2 −𝑉2 be regular splittings of 𝐴1 𝜌(𝑈†𝑉 )𝑦𝑇𝐴† 𝑥=𝑦𝑇𝑈†𝑉 𝐴† 𝑥 𝐴 𝐴−1𝑉 ≥𝐴−1𝑉 2 2 2 2 2 2 and 2,respectively.If 2 2 1 1,then (16) 𝑇 † † † 𝑇 † −1 −1 ≥𝑦 𝐴1𝑉1𝑈1 𝑥=𝜌(𝑈1 𝑉1)𝑦 𝐴1𝑥. 𝜌(𝑈1 𝑉1)≤𝜌(𝑈2 𝑉2)<1. (10) 𝐴† ≥𝐴† Corollary 11 (see [5, Theorem 3.32]). Let 𝐴 be a monotone By assumption 1 2 we obtain matrix and let 𝐴=𝑈1 −𝑉1 =𝑈2 −𝑉2 be two regular splittings 𝜌(𝑈†𝑉 )𝑦𝑇𝐴† 𝑥≥𝜌(𝑈†𝑉 )𝑦𝑇𝐴† 𝑥. of 𝐴.If𝑉2 ≥𝑉1,then 2 2 2 1 1 2 (17)

−1 −1 𝜌(𝑈1 𝑉1)≤𝜌(𝑈2 𝑉2)<1. (11) Therefore † † Next the comparison results for proper weak regular 𝜌(𝑈1 𝑉1)≤𝜌(𝑈2 𝑉2). (18) splittings are given. The case that 𝐴1 =𝑈1 −𝑉1 is of first type and 𝐴2 =𝑈2 −𝑉2 Theorem 12. Let 𝐴1 and 𝐴2 be two semimonotone matrices isofsecondtypecanbeprovedinasimilarway. and let 𝐴1 =𝑈1 −𝑉1 and 𝐴2 =𝑈2 −𝑉2 be proper weak regular The proof is completed. † splittingsofthesametypesof𝐴1 and 𝐴2,respectively.If𝐴2𝑉2 ≥ † 𝐴1𝑉1 ≥𝑂,then When considering the monotone matrices, the condition 𝐴−1𝑉 ≥𝑂 † † 2 2 for the convergence of weak regular splitting 𝜌(𝑈1 𝑉1)≤𝜌(𝑈2 𝑉2)<1. (12) (weak nonnegative splitting in [11]) is not necessary. Hence we have the following corollary. † † Proof. Note that 𝐴2𝑉2 ≥𝐴1𝑉1 ≥𝑂;fromTheorem 4 we have † Corollary 14. 𝐴 𝐴 𝜌(𝑈𝑖 𝑉𝑖) < 1 (𝑖 = 1, 2). Analogous to the proof of Theorem 6, Let 1 and 2 be two monotone matrices and let 𝐴1 =𝑈1 −𝑉1 and 𝐴2 =𝑈2 −𝑉2 be weak regular splittings the desired comparison results are obtained. −1 of different types of 𝐴1 and 𝐴2, respectively. Assume that 𝐴1 − Theorem 13. 𝐴 𝐴 −1 −1 −1 −1 −1 Let 1 and 2 be two semimonotone matrices 𝐴2 ≥𝑂.If𝑈1 −𝑈2 ≥𝐴1 −𝐴2 ,then and let 𝐴1 =𝑈1 −𝑉1 and 𝐴2 =𝑈2 −𝑉2 be proper weak regular −1 −1 splittings of different types of 𝐴1 and 𝐴2, respectively. Assume 𝜌(𝑈 𝑉 )≤𝜌(𝑈 𝑉 )<1. † † † † † † † 1 1 2 2 (19) that 𝐴1 −𝐴2 ≥𝑂and 𝐴2𝑉2 ≥𝑂.If𝑈1 −𝑈2 ≥𝐴1 −𝐴2,then † † 3. Conclusion 𝜌(𝑈1 𝑉1)≤𝜌(𝑈2 𝑉2)<1. (13) In this paper, a new convergence theorem for proper weak Proof. Since 𝐴2 =𝑈2 −𝑉2 is a proper weak regular splitting † regular splitting of a semimonotone matrix and two com- of semimonotone matrix 𝐴2 and 𝐴2𝑉2 ≥𝑂,itfollowsfrom † parison theorems for proper weak regular and proper weak Theorem 4 that 𝜌(𝑈2 𝑉2)<1.Hence,itsufficestoshowthat † † regular splittings of different semimonotone matrices are 𝜌(𝑈1 𝑉1)≤𝜌(𝑈2 𝑉2). given. The obtained results are improved and/or generalized Assume first that 𝐴1 =𝑈1 −𝑉1 is of second type and 𝐴2 = the previous results. Applying the comparison results to judge 𝑈2 −𝑉2 is of first type. Note that the splittings 𝐴1 =𝑈1 −𝑉1 the efficiency of the preconditioners for rectangular linear † † † and 𝐴2 =𝑈2 −𝑉2 are proper splittings; then, 𝑈𝑖 𝑈𝑖𝐴𝑖 =𝐴𝑖 , system needs further study. 4 Journal of Applied Mathematics

Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments The authors would like to thank two anonymous referees and Dr. Zhiping Xiong of Wuyi University for their valuable comments and suggestions, which improve the presentation of this paper. This work was supported by the National Natural Science Foundation of China (Grant no. 11301290) and the Youth Research Ability Project of Northwest Normal University (Grant no. NWNU-LKQN-13-15).

References

[1] D. Mishra and K. Sivakumar, “Comparison theorems for a subclass of proper splittings of matrices,” Applied Mathematics Letters, vol. 25, no. 12, pp. 2339–2343, 2012. [2] A. Ben-Israel and T. Greville, Generalized Inverses: Theory and Applications, Springer, New York, NY, USA, 2003. [3] G. Wang, Y. Wei, and S. Qiao, Generalized Inverses: Theory and Computations, Science Press, Beijing, China, 2004. [4] A. Berman and R. J. Plemmons, “Cones and iterative methods for best least squares solutions of linear systems,” SIAM Journal on Numerical Analysis,vol.11,pp.145–154,1974. [5]R.S.Varga,Matrix Iterative Analysis,vol.27,Springer,Berlin, Germany, 2000. [6] A. Berman and R. Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM, Philadelphia, Pa, USA, 1994. [7] J.-J. Climent and C. Perea, “Iterative methods for least-square problems based on proper splittings,” Journal of Computational and Applied Mathematics,vol.158,no.1,pp.43–48,2003. [8] L. Jena, D. Mishra, and S. Pani, “Convergence and comparison theorems for single and double decompositions of rectangular matrices,” Calcolo, vol. 51, no. 1, pp. 141–149, 2014. [9] J.-J. Climent, A. Devesa, and C. Perea, “Convergence results for proper splittings,”in Recent Advances in Applied and Theoretical Mathematics,N.Mastorakis,Ed.,pp.39–44,2000. [10] S.-X. Miao and B. Zheng, “A note on double splittings of different monotone matrices,” Calcolo,vol.46,no.4,pp.261– 266, 2009. [11] L. Elsner, A. Frommer, R. Nabben, and D. B. Szyld, “Conditions for strict inequality in comparisons of spectral radii of splittings of different matrices,” Linear Algebra and Its Applications,vol. 363, pp. 65–80, 2003. [12] C.-X. Li, Q.-F. Cui, and S.-L. Wu, “Comparison theorems for single and double splittings of matrices,” Journal of Applied Mathematics,vol.2013,ArticleID827826,4pages,2013. [13] D. Mishra, “Nonnegative splittings for rectangular matrices,” Computers & Mathematics with Applications,vol.67,no.1,pp. 136–144, 2014. Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2014, Article ID 408457, 6 pages http://dx.doi.org/10.1155/2014/408457

Research Article Some Results on Characterizations of Matrix Partial Orderings

Hongxing Wang and Jin Xu

Department of Mathematics, Huainan Normal University, Anhui 232001, China

Correspondence should be addressed to Jin Xu; xujin [email protected]

Received 19 February 2014; Accepted 21 April 2014; Published 28 May 2014

Academic Editor: Yang Zhang

Copyright © 2014 H. Wang and J. Xu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Some characterizations of the left-star, right-star, and star partial orderings between matrices of the same size are obtained. Based on those results, several characterizations of the star partial ordering between EP matrices are given. At last, one characterization of the sharp partial ordering between group matrices is obtained.

# † 1. Introduction if and only if 𝐴 is a group matrix with 𝐴 =𝐴.Thesymbols 𝑛 𝑛 𝑛×𝑛 C and C standforthesubsetofC consisting of group C𝑚×𝑛 GP EP In this paper we use the following notation. Let be the matrices and EP matrices, respectively (see, e.g., [1, 2]for 𝑚×𝑛 𝐴∈C𝑚×𝑛 𝐴∗ set of complex matrices. For any matrix , , details). R(𝐴) 𝑟(𝐴) 𝑚×𝑛 ,and denote the conjugate transpose, the range, Five matrix partial orderings defined in C are con- 𝐴 𝐼 𝑛× and the rank of ,respectively.Thesymbol 𝑛 denotes the sidered in this paper. The first of them is the minus partial 𝑛 identity matrix, and 0 denotes a zero matrix of appropriate 𝑚×𝑛 ordering defined by Hartwig [3]andNambooripad[4] size. The Moore-Penrose inverse of a matrix 𝐴∈C , † 𝑛×𝑚 independently in 1980: denoted by 𝐴 , is defined to be the unique matrix 𝑋∈C − − = = satisfying the four matrix equations 𝐴≤𝐵⇐⇒𝐴 𝐴=𝐴 𝐵, 𝐴𝐴 =𝐵𝐴 , (3)

− = (1) 𝐴𝑋𝐴 =𝐴, where 𝐴 ,𝐴 ∈𝐴{1}.In[3]itwasshownthat

(2) 𝑋𝐴𝑋=𝑋, 𝐴≤𝐵⇐⇒𝑟(𝐵−𝐴) =𝑟(𝐵) −𝑟(𝐴) . (4) ∗ (1) (3)(𝐴𝑋) =𝐴𝑋, The rank equality indicates why the minus partial ordering (4)(𝑋𝐴)∗ =𝑋𝐴, is also called the rank-subtractivity partial ordering. In the same paper [3]itwasalsoshownthat − and 𝐴 denotes any solution to the matrix equation 𝐴𝑋𝐴 = ∗ 𝐴 − − 𝐴 ≤𝐵⇐⇒𝑟[ ]=𝑟[𝐴𝐵]=𝑟(𝐵) ,𝐴𝐵𝐴=𝐴, 𝐴 with respect to 𝑋; 𝐴{1} denotes the set of 𝐴 ;thatis,𝐴{1} = 𝐵 (5) # {𝑋 | 𝐴𝑋𝐴.Moreover, =𝐴} 𝐴 denotes the group inverse of 2 − 𝐴 with 𝑟(𝐴 )=𝑟(𝐴),thatis,theuniquesolutionto where 𝐵 ∈𝐵{1}. The second partial ordering of interest is the star partial (1) 𝐴𝑋𝐴 =𝐴, ordering introduced by Drazin [5], which is determined by

(2) 𝑋𝐴𝑋 =𝑋, (2) ∗ † † † † 𝐴 ≤𝐵⇐⇒𝐴𝐴=𝐴𝐵, 𝐴𝐴 =𝐵𝐴. (6) (5) 𝐴𝑋 = 𝑋𝐴. It is well known that # 2 It is well known that 𝐴 exists if and only if 𝑟(𝐴 )=𝑟(𝐴), ∗ 𝐴 ≤𝐵⇐⇒𝐴∗𝐴=𝐴∗𝐵, 𝐴𝐴∗ =𝐵𝐴∗. (7) where case 𝐴 is also called a group matrix. A matrix 𝐴 is EP 2 Journal of Applied Mathematics

In 1991, Baksalary and Mitra [6] defined the left-star and (ii) right-star partial orderings characterized as 𝐵†𝐵𝐴†𝐴 𝐴 ∗≤𝐵⇐⇒𝐴∗𝐴=𝐴∗𝐵, R (𝐴) ⊆ R (𝐵) , 𝐴≤∗𝐵⇐⇒𝑟[ ]=𝑟(𝐵) ; 𝐵𝐴 (16) (8) 𝐴≤∗𝐵⇐⇒𝐴𝐴∗ =𝐵𝐴∗, R (𝐴∗)⊆R (𝐵∗).

The last partial ordering we will deal with in this paper (iii) is the sharp partial ordering, introduced by Mitra [7]in1987, 𝑛 andisdefinedinthesetC by † † GP ∗ 𝐵 𝐵𝐴𝐴 [ ∗ ∗ ] # # # # # 𝐴 ≤𝐵⇐⇒𝑟 𝐵 𝐵𝐴𝐴 =𝑟(𝐵) ; (17) 𝐴≤ 𝐵⇐⇒𝐴𝐴=𝐴𝐵, 𝐴𝐴 =𝐵𝐴. (9) [ 𝐵𝐴] A detailed discussion of partial orderings and their applications can be found in [1, 8–10]. Itiswellknownthatrankofmatrixisanimportanttool (iv) in matrix theory and its applications, and many problems † † are closely related with the ranks of some matrix expressions ∗ 𝐵𝐵 𝐴𝐴 [ ∗ ∗] under some restrictions (see [11–15] for details). Our aim in 𝐴 ≤𝐵⇐⇒𝑟 𝐵𝐵 𝐴𝐴 =𝑟(𝐵) . (18) ∗ ∗ this paper is to characterize the left-star, right-star, star, and [ 𝐵 𝐴 ] sharp partial orderings by applying rank equalities. In the following, when 𝐴 is considered below 𝐵 with respect to one partial ordering, then the partial ordering should entail the Proof. From assumption 𝑟(𝐴) > 𝑟(𝐵). ≥1 𝐵𝐵† 𝐴𝐴† 𝐵𝐵† 𝐴𝐴† 𝐵0 𝑟[ ]≥𝑟([ ][ ]) 2. The Star Partial Ordering 𝐵∗ 𝐴∗ 𝐵∗ 𝐴∗ 0𝐴 𝐴 𝐵 𝑚×𝑛 𝑎 𝐵𝐴 Let and be complex matrices with ranks and =𝑟[ ] ∗ 𝐵∗𝐵𝐴∗𝐴 𝑏,respectively.Let𝐴 ≤𝐵. Then there exist unitary matrices 𝑈∈C𝑚×𝑚 𝑉∈C𝑛×𝑛 (19) and such that 𝐵𝐴𝐵† 0 ≥𝑟([ ∗ ∗ ][ †]) 𝐷𝑎 00 𝐵 𝐵𝐴𝐴 0𝐴 ∗ 𝐷 0 ∗ 𝑈 𝐴𝑉=( 𝑎 ), 𝑈 𝐵𝑉=(0𝐷0), (10) 00 𝐵𝐵† 𝐴𝐴† 000 =𝑟[ ], 𝐵∗ 𝐴∗ where both the 𝑎×𝑎matrix 𝐷𝑎 and the (𝑏−𝑎)×(𝑏−𝑎) matrix 𝐷 are real, diagonal, and positive definite (see [16,Theorem we have 2]). In [1, Theorem 5.2.8], it was also shown that ∗ 𝐵∗𝐵𝐴∗𝐴 𝐵𝐵† 𝐴𝐴† 𝐴 ≤𝐵⇐⇒𝐴†𝐴=𝐵†𝐴, 𝐴𝐴† =𝐴𝐵†. (11) 𝑟[ ]=𝑟[ ]. 𝐵𝐴 𝐵∗ 𝐴∗ (20) In [17], Wang obtained the following characterizations of the left-star and right-star partial orderings for matrices: Applying (12) gives (i). 𝐵∗𝐵𝐴∗𝐴 Inthesameway,applying 𝐴 ∗≤𝐵⇐⇒𝑟[ ]=𝑟(𝐵) , 𝐵𝐴 (12) 𝐵†𝐵𝐴†𝐴 𝐵𝐵∗ 𝐴𝐴∗ 𝑟[ ]=𝑟[ ] 𝐵𝐵∗ 𝐴𝐴∗ 𝐵𝐴 𝐵∗ 𝐴∗ (21) 𝐴≤∗𝐵⇐⇒𝑟[ ]=𝑟(𝐵) , 𝐵∗ 𝐴∗ (13)

∗ 𝐵∗𝐵𝐴∗𝐴 and (13) gives (ii). 𝐴 ≤𝐵⇐⇒𝑟[ ]=𝑟(𝐵) , 𝐵𝐴 If (14) † † 𝐵𝐵∗ 𝐴𝐴∗ 𝐵 𝐵𝐴𝐴 𝑟[ ]=𝑟(𝐵) . [ ∗ ∗ ] 𝐵∗ 𝐴∗ 𝑟 𝐵 𝐵𝐴𝐴 =𝑟(𝐵) , (22) [ 𝐵𝐴] 𝑚×𝑛 Theorem 1. Let 𝐴, 𝐵 ∈ C .Then (i) then 𝐵𝐵† 𝐴𝐴† 𝐵𝐵∗ 𝐴𝐴∗ 𝐵𝐵† 𝐴𝐴† 𝐴 ∗≤𝐵⇐⇒𝑟[ ]=𝑟(𝐵) ; 𝑟[ ]=𝑟(𝐵) ,𝑟[ ]=𝑟(𝐵) . 𝐵∗ 𝐴∗ (15) 𝐵∗ 𝐴∗ 𝐵∗ 𝐴∗ (23) Journal of Applied Mathematics 3

∗ Theorem 2. 𝐴, 𝐵 ∈ C𝑛 𝑟(𝐵) ≥ 𝑟(𝐴) Applying (i), (ii), and (14), we obtain 𝐴 ≤𝐵.Conversely,if Let EP, .Then ∗ † † 𝐴 ≤𝐵,byusing(11)and(14), we have 𝐴 𝐴−𝐵 𝐴=0,and (v) 0𝐴†𝐴−𝐵†𝐴 𝐵𝐴 𝐵∗𝐵𝐴∗𝐴 𝐴≤∗𝐵⇐⇒𝑟[ ]=𝑟(𝐵) ; 𝑟[ ]=𝑟[𝐵∗𝐵𝐴∗𝐴 ] 2 2 (28) 𝐵𝐴 𝐵 𝐴 [ 𝐵𝐴]

† † † (vi) 𝐼𝑛 0𝐵 0𝐴𝐴−𝐵 𝐴 [ ] [ ∗ ∗ ] 2 =𝑟( 0𝐼𝑛 0 𝐵 𝐵𝐴𝐴 ) 𝐵𝐵 𝐴 ∗≤𝐵⇐⇒𝑟[ ]=𝑟(𝐵) . [ 00𝐼𝑚] [ 𝐵𝐴] 𝐴𝐴2 (29) 𝐵†𝐵𝐴†𝐴 ∗ ∗ 𝑛 † † =𝑟[𝐵 𝐵𝐴𝐴] , Proof. By 𝐴, 𝐵 ∈ CEP,itisobviousthat𝐴𝐴 =𝐴𝐴 and † † [ 𝐵𝐴] 𝐵𝐵 =𝐵𝐵.Then † † 𝐵†𝐵𝐴†𝐴 𝐵𝐴 𝐵 𝐵𝐴𝐴 𝑟[ 2 2]=𝑟(𝐵) ⇐⇒ 𝑟 [ ]=𝑟(𝐵) . (30) 𝑟 (𝐵) =𝑟[𝐵∗𝐵𝐴∗𝐴] . 𝐵 𝐴 𝐵𝐴 [ 𝐵𝐴] (24) Hence, we have (v). The proof of (vi) is similar to that of (v).

Hence, we have (iii). 𝑛 ∗ † Theorem 3. Let 𝐴, 𝐵 ∈ CEP.Then Similarly, applying 𝐴 ≤𝐵,(11), and (14), we obtain 𝐴𝐴 − † † † ∗ ∗ † ∗ 𝐴𝐵 =0, 𝐴𝐵 =(𝐴𝐵) =(𝐵 ) 𝐴 ,and (vii) † † † † ∗ ∗ 0𝐴𝐴−𝐴𝐵 ∗ 𝐵𝐵 𝐴𝐴 𝐵𝐵 𝐴𝐴 ∗ ∗ [ ] 𝑟[ ∗ ∗ ]=𝑟[𝐵𝐵 𝐴𝐴 ] 𝐴 ≤𝐵⇐⇒𝑟 𝐵𝐴=𝑟(𝐵) ; (31) 𝐵 𝐴 ∗ ∗ ∗ ∗ [ 𝐵 𝐴 ] [ 𝐵 𝐴 ] ∗ † 0𝐴𝐴† −𝐴𝐵† 𝐼𝑚 0(𝐵) (viii) [ ] [ ∗ ∗ ] =𝑟( 0𝐼𝑛 0 𝐵𝐵 𝐴𝐴 ) ∗ ∗ 𝐵∗𝐵𝐴∗𝐴 [ 00 𝐼𝑛 ] [ 𝐵 𝐴 ] ∗ 𝐴 ≤𝐵⇐⇒𝑟[ 𝐵𝐴] =𝑟(𝐵) ; (32) ∗ ∗ =𝑟(𝐵) . [ 𝐵 𝐴 ] (25) (ix) Then, we obtain (iv). 𝐵𝐵𝐴 In [9,Theorem2.1],Ben´ıtez et al. deduce the character- ∗ 𝐴 ≤𝐵⇐⇒𝑟[𝐵𝐴𝐵] =𝑟(𝐵) ; (33) izations of the left-star, right-star, and star partial orderings [𝐴𝐴𝐵] for matrices, when at least one of the two involved matrices 𝑛×𝑛 𝑛×𝑛 is EP. When both 𝐴∈C and 𝐵∈C are EP matrices, [1, Theorems 5.4.15 and 5.4.2] give the following results: (x) † ∗ 𝐵𝐵𝐴 ∗ ∗ ∗ 𝐴 ≤ 𝐵 ⇐⇒ 𝐴 ≤ 𝐵, 𝐴𝐵 𝐵 𝐴 . † and are Hermitian 𝐴 ≤𝐵⇐⇒𝑟[𝐵𝐴𝐵] =𝑟(𝐵) ; (34) (26) † ∗ 2 [𝐴𝐴𝐵] 𝐴 ≤𝐵⇐⇒(𝐴𝐵)† =𝐵†𝐴† =𝐴†𝐵† =𝐴† .

∗ (xi) In addition, it was also shown that 𝐴 ≤𝐵if and only if 𝐴 and 𝐵 have the form 𝐵𝐵𝐴∗ ∗ ∗ 𝐴 ≤𝐵⇐⇒𝑟[𝐵𝐴𝐵] =𝑟(𝐵) . (35) 𝑇00 𝑇00 ∗ ∗ ∗ [𝐴𝐴𝐵] 𝐴=𝑈[000] 𝑈 ,𝐵=𝑈[0𝐾0] 𝑈 , (27) 000 000 [ ] [ ] 𝑛 † † Proof. By 𝐴, 𝐵 ∈ CEP,itisobviousthat𝐴𝐴 =𝐴𝐴 and 𝑟(𝐴)×𝑟(𝐴) (𝑟(𝐵)−𝑟(𝐴))×(𝑟(𝐵)−𝑟(𝐴)) † † where 𝑇∈C is nonsingular, 𝐾∈C 𝐵𝐵 =𝐵𝐵. Applying (i), (ii), and the rank equality in (vii) 𝑛×𝑛 is nonsingular, and 𝑈∈C is unitary (see [1,Theorem we obtain 5.4.1]). 𝐵†𝐵𝐴†𝐴 𝐵𝐵† 𝐴𝐴† Based on these results, we consider the characterizations 𝑟[ ]=𝑟(𝐵) ,𝑟[ ]=𝑟(𝐵) ; 𝑛 𝐵𝐴 𝐵∗ 𝐴∗ (36) of the star partial ordering for matrices in the set of CEP. 4 Journal of Applied Mathematics

∗ ∗ −1 that is, 𝐴 ≤𝐵.Conversely,supposethat𝐴 ≤𝐵. Applying Correspondingly denote 𝑃 𝐵𝑃 by 𝐴−𝐴𝐴†𝐵=0 𝐵∗𝐵𝐵† =𝐵∗ and ,weobtain 𝐵 𝐵 𝐵=𝑈 [ 1 2]𝑈∗, 1 𝐵 𝐵 1 (42) 𝐵𝐵† 𝐴𝐴† 3 4 𝐵𝐵† 𝐴𝐴† 𝑟 (𝐵) =𝑟[ ]=𝑟[ 𝐵𝐴] 𝑟(𝐴)×𝑟(𝐴) 𝐵𝐴 ∗ ∗ † where 𝐵1 ∈ C .Itfollowsthat [ 0𝐴−𝐵 𝐴𝐴 ] (37) 𝑇𝐵 𝑇𝐵 𝐵 𝑇0 𝑇𝐵 𝑇𝐵 𝑇2 0 † † [ 1 2]=[ 1 ], [ 1 2]=[ ]. 𝐵𝐵 𝐴𝐴 00 𝐵 𝑇0 00 00 =𝑟[ 𝐵𝐴] . 3 ∗ ∗ (43) [ 𝐵 𝐴 ] Since 𝑇 is a unitary matrix, ∗ † ∗ ∗ † Applying (11), we obtain 𝐵 𝐵𝐵 𝐵=𝐵𝐵 and 𝐵 𝐵𝐴 𝐴= ∗ ∗ † ∗ † ∗ † ∗ † 𝐵 =𝑇, 𝐵 =0, 𝐵 =0. 𝐴 𝐴 and also (𝐵 𝐵) 𝐵 𝐵=𝐵𝐵 and (𝐵 𝐵) 𝐴 𝐴=𝐴𝐴. 1 2 3 (44) Then Thus 𝐵𝐵† 𝐴𝐴† 𝐵†𝐵𝐴†𝐴 𝑇0 𝐵=𝑈[ ]𝑈∗. 𝑟 [ 𝐵𝐴] =𝑟[ 𝐵𝐴] 0𝐵 (45) ∗ ∗ ∗ ∗ 4 [ 𝐵 𝐴 ] [ 𝐵 𝐴 ] Since 𝐵 is EP, 𝐵4 is EP, and there exists a unitary matrix 𝐵∗𝐵0 0 𝐵†𝐵𝐴†𝐴 (𝑛−𝑟(𝐴))×(𝑛−𝑟(𝐴)) 𝑈2 ∈ C and a nonsingular matrix 𝐾∈ [ ] [ ] (𝑟(𝐵)−𝑟(𝐴))×(𝑟(𝐵)−𝑟(𝐴)) ≥𝑟( 0𝐼𝑛 0 𝐵𝐴) C ∗ ∗ such that [ 00𝐼𝑛] [ 𝐵 𝐴 ] 𝐾0 𝐵 =𝑈 [ ]𝑈∗. 𝐵∗𝐵𝐴∗𝐴 4 2 00 2 (46) =𝑟[ 𝐵𝐴] ∗ ∗ [ 𝐵 𝐴 ] Write 00 ∗ † ∗ ∗ 𝑈=𝑈 [ ]. (𝐵 𝐵) 00 𝐵 𝐵𝐴𝐴 1 0𝑈 (47) [ ] [ ] 2 ≥𝑟( 0𝐼𝑛 0 𝐵𝐴) ∗ ∗ [ 00𝐼𝑛] [ 𝐵 𝐴 ] Then 𝐴 and 𝐵 have the form 𝐵†𝐵𝐴†𝐴 𝑇00 𝑇00 [ ] ∗ [ ] ∗ =𝑟[ 𝐵𝐴] ; 𝐴=𝑈 000 𝑈 ,𝐵=𝑈0𝐾0 𝑈 . (48) ∗ ∗ 000 000 [ 𝐵 𝐴 ] [ ] [ ] (38) ∗ Applying (27), we have 𝐴 ≤𝐵. that is, The proofs of (x) and (xi) are similar to that of (ix). † † 𝐵∗𝐵𝐴∗𝐴 𝐵𝐵 𝐴𝐴 3. The Sharp Partial Ordering 𝑟 [ 𝐵𝐴] =𝑟[ 𝐵𝐴] . (39) 𝐵∗ 𝐴∗ 𝐵∗ 𝐴∗ 𝐴, 𝐵 ∈ C𝑛 𝑎 𝑏 [ ] [ ] Let GP with ranks and ,respectively.Itiswell known that Hence, we have (viii). ∗ # 2 𝑛 𝐴≤ 𝐵⇐⇒𝐴 =𝐴𝐵=𝐵𝐴. (49) Suppose that 𝐴 ≤𝐵.Since𝐴, 𝐵 ∈ CEP, applying (27), it is easy to check the rank equality in (ix). Conversely, under the # In addition, 𝐴≤ 𝐵 if and only if 𝐴 and 𝐵 can be written as rank equality in (ix), we have 𝐸00 𝐸00 𝐵𝐵𝐴 𝐵𝐵𝐴 −1 −1 𝑟[ ]=𝑟[ ]=𝑟(𝐵) 󳨐⇒ 𝐴𝐵 =𝐵𝐴, 𝐴=𝑃[000] 𝑃 ,𝐵=𝑃[0𝐸󸀠 0] 𝑃 , 𝐵𝐴𝐵 0𝐴𝐵−𝐵𝐴 (50) [000] [000]

𝐵𝐵𝐴 𝐵0 2 𝑎×𝑎 󸀠 (𝑏−𝑎)×(𝑏−𝑎) 𝑟[ ]=𝑟[ 2]=𝑟(𝐵) 󳨐⇒ 𝐴 𝐵 =𝐴 . where 𝐸∈C is nonsingular, 𝐸 ∈ C is 𝐴𝐴𝐵 𝐴𝐴𝐵−𝐴𝐴 𝑛×𝑛 nonsingular, and 𝑃∈C is nonsingular (see [18]). (40) In Theorem 4, we give one characterization of the sharp 𝑛×𝑛 partial ordering by using one rank equality. Since 𝐴 is EP, there exists a unitary matrix 𝑈1 ∈ C and a 𝑇∈C𝑟(𝐴)×𝑟(𝐴) 𝑛 nonsingular matrix such that Theorem 4. Let 𝐴, 𝐵 ∈ CGP.Then 𝑇0 𝐴𝐵𝐴 𝐴=𝑈 [ ]𝑈∗. 𝐴≤#𝐵⇐⇒𝑟[ ]=𝑟(𝐴𝐵𝐴) . 1 00 1 (41) 𝐴𝐵 𝐴𝐵𝐴 (51) Journal of Applied Mathematics 5

Proof. Let 𝐴 have the core-nilpotent decomposition (see [19, Conflict of Interests Exercise 5.10.12]) The authors declare that there is no conflict of interests Σ0 𝐴=𝑃[ ]𝑃−1, regarding the publication of this paper. 00 (52)

𝑟(𝐴)×𝑟(𝐴) 𝑛×𝑛 Acknowledgments with nonsingular matrices Σ∈C and 𝑃∈C . −1 Correspondingly denote 𝑃 𝐵𝑃 by The authors would like to thank the referees for their helpful comments and suggestions. The work of the first author was −1 𝐵1 𝐵2 supported in part by the Foundation of Anhui Educational 𝑃 𝐵𝑃 =[ ], (53) 𝐵3 𝐵4 Committee (Grant no. KJ2012B175) and the National Natural Science Foundation of China (Grant no. 11301529). The work 𝑟(𝐴)×𝑟(𝐴) where 𝐵1 ∈ C .Itfollowsthat of the second author was supported in part by the Foundation of Anhui Educational Committee (Grant no. KJ2013B256). 𝑟 (𝐴𝐵𝐴) =𝑟(Σ𝐵1Σ) , Σ0𝐵Σ References 𝐴𝐵𝐴 1 𝑟[ ]=𝑟[ 00𝐵Σ ] 𝐴𝐵 𝐴𝐵𝐴 3 Σ𝐵 Σ𝐵 Σ𝐵 Σ [1] S. K. Mitra, P. Bhimasankaram, and S. B. Malik, Matrix Partial [ 1 2 1 ] Orders, Shorted Operators and Applications, World Scientific, Σ0 0 Singapore, 2010. [ ] [2] G. Wang, Y. Wei, and S. Qiao, Generalized Inverses: Theory and =𝑟 00 𝐵3Σ 0Σ𝐵 Σ𝐵 Σ−Σ𝐵 Σ−1𝐵 Σ Computations, Science Press, Beijing, China, 2004. [ 2 1 1 1 ] [3]R.E.Hartwig,“Howtopartiallyorderregularelements,” 0𝐵Σ Mathematica Japonica,vol.25,no.1,pp.1–13,1980. =𝑟(Σ) +𝑟[ 3 ]. Σ𝐵 Σ𝐵 Σ−Σ𝐵 Σ−1𝐵 Σ [4]K.S.S.Nambooripad,“Thenaturalpartialorderonaregular 2 1 1 1 semigroup,” Proceedings of the Edinburgh Mathematical Society, (54) vol.23,no.3,pp.249–260,1980. [5] M. P. Drazin, “Natural structures on semigroups with involu- Applying (54) to the rank equality in (51), we obtain tion,” Bulletin of the American Mathematical Society,vol.84,no. 0𝐵Σ 1, pp. 139–141, 1978. 𝑟[ 3 ]+𝑟(Σ) =𝑟(Σ𝐵Σ) . Σ𝐵 Σ𝐵 Σ−Σ𝐵 Σ−1𝐵 Σ 1 (55) [6] J. K. Baksalary and S. K. Mitra, “Left-star and right-star partial 2 1 1 1 orderings,” Linear Algebra and Its Applications,vol.149,pp.73– 89, 1991. Hence 𝑟(Σ𝐵1Σ) = 𝑟(Σ), Σ𝐵2 =0, 𝐵3Σ=0,andΣ𝐵1Σ= Σ𝐵 Σ−1𝐵 Σ Σ∈C𝑟(𝐴)×𝑟(𝐴) 𝐵 ∈ [7]S.K.Mitra,“Ongroupinversesandthesharporder,”Linear 1 1 .Since is invertible and 1 Algebra and Its Applications,vol.92,pp.17–37,1987. C𝑟(𝐴)×𝑟(𝐴) ,itfollowsimmediatelythat [8]J.K.Baksalary,O.M.Baksalary,andX.Liu,“Furtherproperties of the star, left-star, right-star, and minus partial orderings,” 𝑟(𝐵 )=𝑟(Σ) ,𝐵=0, 𝐵 =0, 𝐵 =Σ. 1 3 2 1 (56) Linear Algebra and Its Applications,vol.375,pp.83–94,2003. [9] J. Ben´ıtez, X. Liu, and J. Zhong, “Some results on matrix partial Therefore orderings and reverse order law,” Electronic Journal of Linear Algebra,vol.20,pp.254–273,2010. Σ0 −1 𝐵=𝑃[ ]𝑃 . (57) [10] J. Groß, “Remarks on the sharp partial order and the ordering 0𝐵4 of squares of matrices,” Linear Algebra and Its Applications,vol. Applying 417, no. 1, pp. 87–93, 2006. [11] Z.-J. Bai and Z.-Z. Bai, “On nonsingularity of block two-by-two Σ2 0 matrices,” Linear Algebra and Its Applications,vol.439,no.8,pp. 𝐴2 =𝑃[ ]𝑃−1 00 2388–2404, 2013. [12] D. Chu, Y. S. Hung, and H. J. Woerdeman, “Inertia and rank Σ0 Σ0 characterizations of some matrix expressions,” SIAM Journal on =𝑃[ ]𝑃−1𝑃[ ]𝑃−1 00 0𝐵4 Matrix Analysis and Applications,vol.31,no.3,pp.1187–1226, 2009. (58) =𝐴𝐵 [13] Y. Liu and Y. Tian, “A simultaneous decomposition of a matrix triplet with applications,” Numerical Linear Algebra with Σ0 Σ0 =𝑃[ ]𝑃−1𝑃[ ]𝑃−1 Applications,vol.18,no.1,pp.69–85,2011. 0𝐵4 00 [14] H. Wang, “The minimal rank of 𝐴−𝐵𝑋with respect to Hermitian matrix,” Applied Mathematics and Computation,vol. =𝐵𝐴, 233, pp. 55–61, 2014.

# [15] Q.-W. Wang and Z.-H. He, “Solvability conditions and general and (49), we obtain that 𝐴≤ 𝐵. solution for mixed Sylvester equations,” Automatica,vol.49,no. Conversely, it is a simple matter. 9,pp.2713–2719,2013. 6 Journal of Applied Mathematics

[16] R. E. Hartwig and G. P.H. Styan, “On some characterizations of the “star” partial ordering for matrices and rank subtractivity,” Linear Algebra and Its Applications,vol.82,pp.145–161,1986. [17] H. X. Wang, “Rank characterizations of some matrix partial orderings,” Journal of East China Normal University,no.5,pp. 5–11, 2011. [18] Z.J.WangandX.J.Liu,“Onthreepartialorderingsofmatrices,” Journal of Mathematical Study,vol.36,no.1,pp.75–81,2003. [19] C. D. Meyer, Matrix Analysis and Applied Linear Algebra, Society for Industrial and Applied Mathematics, Philadelphia, Pa,USA,2000. Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2014, Article ID 515082, 6 pages http://dx.doi.org/10.1155/2014/515082

Research Article A Test Matrix for an Inverse Eigenvalue Problem

G. M. L. Gladwell,1 T. H. Jones,2 and N. B. Willms2

1 Department of Civil and Environmental Engineering, University of Waterloo, Waterloo, ON, Canada N2L 3G1 2 Department of Mathematics, Bishop’s University, Sherbrooke, QC, Canada J1M 2H2

Correspondence should be addressed to N. B. Willms; [email protected]

Received 21 February 2014; Accepted 30 April 2014; Published 26 May 2014

Academic Editor: K. C. Sivakumar

Copyright © 2014 G. M. L. Gladwell et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

𝑛−1 We present a real symmetric tridiagonal matrix of order 𝑛 whose eigenvalues are {2𝑘}𝑘=0 which also satisfies the additional condition 𝑛−2 that its leading principle submatrix has a uniformly interlaced spectrum, {2𝑙 + 1}𝑙=0 . The matrix entries are explicit functions of the size 𝑛, and so the matrix can be used as a test matrix for eigenproblems, both forward and inverse. An explicit solution of a spring- mass inverse problem incorporating the test matrix is provided.

1. Introduction issues.Lossofsignificantfiguresduetoaccumulationof round-off error makes some of the known solution pro- Wearemotivatedbythefollowinginverseeigenvalueprob- cedures undesirable. Determining the extent of round-off lemfirststudiedbyHochstadtin1967[1]. Given two strictly error in the numerical solution, 𝐵̂,computedfromagiven interlaced sequences of real values, dataset requires aprioriknowledge of the exact solution 𝐵. 𝑛 𝑜 𝑛−1 In the absence of this knowledge, an additional numerical (𝜆 ) ,(𝜆) , (1) 𝑖 1 𝑖 1 computation of the forward problem to find the spectra 𝜆(𝐵)̂ 𝜆(𝐵̂𝑜) with and allows comparison to the original data. Test matrices, with known entries and known spectra, 𝑜 𝑜 𝑜 𝜆1 <𝜆1 <𝜆2 <𝜆2 <⋅⋅⋅<𝜆𝑛−1 <𝜆𝑛−1 <𝜆𝑛, (2) are therefore helpful in comparing the efficacy of the various solution algorithms in regard to stability. It is particularly find the 𝑛×𝑛, real, symmetric, and tridiagonal matrix, 𝐵, 𝑛 𝑜 helpful when test matrices can be produced at arbitrary size. such that 𝜆(𝐵) = (𝜆𝑖)1 are the eigenvalues of 𝐵, while 𝜆(𝐵 )= However some existent test matrices given as a function of 𝑜 𝑛−1 𝑛 (𝜆𝑖 )1 are the eigenvalues of the leading principal submatrix matrix size suffer the following trait: when ordered by 𝑜 of 𝐵,where𝐵 is obtained from 𝐵 by deleting the last row and size, the minimum spacing between consecutive eigenval- column. The condition on the dataset (2)isbothnecessary uesisadecreasingfunctionof𝑛. This trait is potentially and sufficient for the existence of a unique Jacobian matrix undesirable since the reciprocal of this minimum separation solution to the problem (see [2], Section 4.3 or [3], Section 1.2 between eigenvalues can be thought of as a condition number for a history of the problem and Section 3 of this paper for on the sensitivity of the eigenvectors (invariant subspaces) additional background theory). to perturbation (see [10], Theorem 8.1.12). Some of the Anumberofdifferentconstructiveprocedurestoproduce algorithms for the inverse problem seem to suffer from the exact solution of this inverse problem have been devel- this form of ill-conditioning. From a motivation to avoid oped [4–9], but none provide an explicit characterization of confounding the numerical stability issue with potential the entries of the solution matrix, 𝐵, in terms of the dataset increased ill-conditioning of the dataset as a function of 𝑛, (2). Computer implementation of these procedures intro- the authors developed a test matrix which has equally spaced duces floating point error and associated numerical stability and uniformly interlaced simple eigenvalues. 2 Journal of Applied Mathematics

In Section 2 we provide the explicit entries of such a Now we show that 𝐴(𝑛 + 1) has eigenvalues {2𝑛} ∪ matrix, 𝐴(𝑛). We claim that its eigenvalues are equally spaced {eigenvalues of 𝐴(𝑛)}.Let𝐶 = 𝐴(𝑛+1)−2𝑛𝐼.Factorize 𝑇 as 𝐶=−𝐿𝐿,where𝐿 is lower bidiagonal. We find

𝜆 (𝐴 (𝑛)) = {0, 2, 4, . . . , 2𝑛 −} 2 , 2𝑛−𝑖+1 𝑖 (3) 𝑙 = √ ;𝑙 =−√ , 𝑖=1,2,...,𝑛−1, 𝑖𝑖 2 𝑖+1,𝑖 2 𝑜 while its leading principal submatrix 𝐴 (𝑛) has eigenvalues (9) 𝑛+1 uniformly interlaced with those of 𝐴(𝑛),namely, 𝑙 = √ ;𝑙=−√𝑛; 𝑙 =0. 𝑛𝑛 2 𝑛+1,𝑛 𝑛+1,𝑛+1 𝜆(𝐴𝑜 (𝑛))={1,3,5,...,2𝑛−3} . (4) Therefore 𝐶 has eigenvalue 0 and thus 𝐴(𝑛+1)has eigenvalue 2𝑛. 𝑇 A short proof verifies the claims. In Section 3 we present Define 𝐷=2𝑛𝐼−𝐿 𝐿;so some background theory concerning Jacobian matrices, and 𝑜 in Section 4 we apply our test matrix to a model of a physical 𝐷 𝑂 𝐷=[ ] (10) spring-mass system, an application which leads naturally to 𝑂2𝑛 Jacobian matrices. with 2𝑛 − 1 1 2. Main Result 𝑑 = ;𝑑 = √𝑖 (2𝑛 − 𝑖), 𝑖=1,2,...,𝑛−1, 𝑖𝑖 2 𝑖+1,𝑖 2 Let 𝐴(𝑛) be an 𝑛×𝑛real symmetric tridiagonal matrix with 𝑛−1 entries 𝑑 = . 𝑛𝑛 2

𝑎𝑖𝑖 = 𝑛−1, 𝑖=1,2,...,𝑛 (11) 𝑜 1 Now 𝐷 hasthesameeigenvaluesas𝐴(𝑛) since they are 𝑎 = √𝑖 (2𝑛−𝑖−1), 𝑖=1,2,...,𝑛−2 𝑜 𝑖,𝑖+1 2 (5) similar matrices via 𝑆𝐷 = 𝐴(𝑛)𝑆 where 𝑆 is upper triangular with entries √ 𝑛 (𝑛−1) 𝑎𝑛−1,𝑛 = √ √ 2 𝑠𝑖𝑖 = 2𝑛 − 𝑖;𝑖,𝑖+1 𝑠 =− 𝑖, 𝑖=1,2,...,𝑛−1, (12) 𝑜 √ and let 𝐴 (𝑛) be the principal submatrix of 𝐴(𝑛),thatis,the 𝑠𝑛𝑛 = 2𝑛;𝑖𝑗 𝑠 =0, otherwise. (𝑛 − 1) × (𝑛 −1) matrix obtained from 𝐴(𝑛) by deleting the last row and column. Therefore 𝐴(𝑛 + 1) has eigenvalues {2𝑛} ∪eigenvalues { of 𝐴(𝑛)}. 𝑜 Theorem 1. 𝐴(𝑛) has eigenvalues {0,2,...,2𝑛−2}and 𝐴 (𝑛) {1,3,...,2𝑛−3} has eigenvalues . 3. Discussion 𝑛=2 Proof. By induction, when Areal,symmetric𝑛×𝑛tridiagonal matrix 𝐵 is called a Jacobian matrix when its off-diagonal elements are nonzero 11 𝐴 (2) =[ ] ([2], page 46). We write 11 (6)

𝑎1 −𝑏1 0 0 ⋅⋅⋅ 0 𝑜 [−𝑏 𝑎 −𝑏 0 ⋅⋅⋅ 0 ] has eigenvalues 0,2, and 𝐴 (2) has eigenvalue 1. Assume the [ 1 2 2 ] [ ] result holds for 𝑛.So𝐴(𝑛) has eigenvalues {0,2,...,2𝑛−2}. [ . ] 𝑜 [ 0−𝑏2 𝑎3 −𝑏3 d . ] Let 𝐵=𝐴(𝑛 + 1) − 𝑛𝐼 and 𝐴=𝐴(𝑛)−(𝑛−1)𝐼.Then𝐵 and 𝐵=[ ] . (13) [ 00dd d 0 ] 𝐴 are similar via 𝐵𝑅 = 𝑅𝐴 where 𝑅 is upper triangular, with [ ] [ . . ] entries . . d −𝑏𝑛−2 𝑎𝑛−1 −𝑏𝑛−1 [ 00⋅⋅⋅0−𝑏𝑛−1 𝑎𝑛 ] { 𝑘 (𝑗 − 1)! (2𝑛 − 𝑗 −1)! {√ ̂ −1 −1 { (𝑖−1)! (2𝑛−𝑖+1)! The similarity transformation, 𝐵=𝑆 𝐵𝑆,where𝑆=𝑆 𝑟𝑖𝑗 = { (7) is the alternating sign matrix, 𝑆=diag(1,−1,1,−1,..., { 𝑖, 𝑗 have same parity and 𝑗≥𝑖, 𝑛−1 { (−1) ), produces a Jacobian matrix 𝐵̂ with entries same as {0 otherwise, 𝐵 except for the sign of the off-diagonal elements, which are all reversed. If instead we use the self-inverse sign matrix, 2𝑗=𝑛,̸ 𝑛−𝑚 𝑘={ (𝑚) ⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ ̂ 1𝑗=𝑛. (8) 𝑆 = diag(1,1,...,1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟, −1,−1,...,−1),totransform𝐵,then𝐵 𝑚 is a Jacobian matrix identical to 𝐵 except for a switched sign 𝑜 Therefore 𝐴 (𝑛 + 1) has eigenvalues {1,3,...,2𝑛−1}. on the 𝑚th off-diagonal element. In regard to the spectrum of Journal of Applied Mathematics 3 the matrix, there is therefore no loss of generality in accepting Jacobian). The inverse problem is also well-posed: there is the convention that a Jacobian matrix is expressed with a unique (up to the signs of the off-diagonal elements) negative off-diagonal elements; that is, 𝑏𝑖 >0,forall𝑖= Jacobian matrix 𝐵 having given spectra specified as per (2) 1,...,𝑛−1 in (13). (see [2], Theorem 4.2.1, noting that the interlaced spectrum 𝑜 𝑛−1 While Cauchy’s interlace theorem [11]guaranteesthat of 𝑛−1eigenvalues (𝜆 )1 canbeusedtocalculatethelast the eigenvalues of any square, real, symmetric (or even components of each of the 𝑛 orthonormalized eigenvectors Hermitian) matrix will interlace those of its leading (or of 𝐵 via equation 4.3.31). Therefore, the matrix 𝐴(𝑛) in trailing) principal submatrix, the interlacing cannot be strict, Theorem 1 istheuniqueJacobianmatrixwitheigenvalues in general [12]. However, specializing to the case of Jacobian equally spaced by two, starting with smallest eigenvalue matrices restricts the interlacing to strict inequalities. That zero, whose leading principal submatrix has eigenvalues also is, Jacobian matrices possess distinct eigenvalues, and the equally spaced by two, starting with smallest eigenvalue one. eigenvalues of the leading (or trailing) principal submatrix As a consequence of the theorem, we now have the are also distinct and strictly interlace those of the original following. matrix (see [2], Theorems 3.1.3 and 3.1.4.Seealso[10] exercise P8.4.1, page 475: when a tridiagonal matrix has Corollary 2. The eigenvalues of the real, symmetric 𝑛×𝑛 algebraically multiple eigenvalues, the matrix fails to be tridiagonal matrix

𝑛−1 [ 𝑎−𝑐√ 0 0 ⋅⋅⋅ 0 ] [ 2 ] [ ] [ 𝑛−1 2𝑛 − 3 ] [−𝑐√ 𝑎−𝑐√ 0 ⋅⋅⋅ 0 ] [ 2 2 ] [ ] [ 2𝑛 − 3 3𝑛 − 6 . ] [ 0−𝑐√ 𝑎−𝑐√ d . ] 𝑊𝑛 = [ 2 2 ] (14) [ ] [ 00dd d0 ] [ ] [ . . (𝑛−2)(𝑛+1) 𝑛 (𝑛−1)] [ . . d −𝑐√ 𝑎−𝑐√ ] [ 4 2 ] [ 𝑛 (𝑛−1) ] 0 0 ⋅⋅⋅ 0 −𝑐√ 𝑎 [ 2 ]

form the arithmetic sequence, of the problem. His proof is now found in [13]. Though longer than the one presented here, his proof utilizes the spectral 𝜆(𝑊)={𝑎 +2𝑐(𝑖−1)}𝑛 , 𝑛 𝑜 𝑖=1 (15) properties of another tridiagonal (nonsymmetric) matrix, the so-called Kac-Sylvester matrix, 𝐾𝑛,ofsize(𝑛+1)×(𝑛+1),with 𝑜 𝑛 while the eigenvalues of its leading principal submatrix, 𝑊𝑛 , eigenvalues 𝜆(𝐾𝑛)={2𝑘−𝑛}𝑘=0 [14–17]: form the uniformly interlaced sequence

𝑜 𝑛−1 𝑛𝑛−1 0 0 ⋅⋅⋅ 0 𝜆(𝑊 )={𝑎𝑜 +𝑐+2𝑐(𝑖−1)} , [ ] 𝑛 𝑖=1 [1𝑛𝑛−20⋅⋅⋅0] (16) [ ] 𝑤ℎ𝑒𝑟𝑒 𝑎= +𝑐(𝑛−1) . [ . ] 𝑜 [02 𝑛𝑛−3d . ] 𝐾𝑛 = [ ] . (17) [00 ddd0] 𝑊 [ ] The form and properties of 𝑛 were first hypothesised [ . . ] by the third author while programming Fortran algorithms . . d 𝑛−2 𝑛 1 to reconstruct band matrices from spectral data [3]. Initial [0 0 ⋅⋅⋅ 0 𝑛−1] 𝑛 attempts to prove the spectral properties of 𝑊𝑛 by both he and his graduate supervisor (the first author) failed. Later, the first Therefereehaspointedouttheconnectionbetween author produced the short induction argument of Theorem 1, the spectra (3)and(4) and the classical orthogonal Hahn in July 1996. Alas, the fax on which the argument was polynomials of a discrete variable [18]. Using (3) as nodes communicated to the third author was lost in a cross-border with weights academic move, and so the matter languished until recently. 𝑛−1 𝑜 In summer of 2013, the second and third authors assigned the ∏𝑗=1 (𝜆𝑖 −𝜆𝑗) 𝜔𝑖−1 = , 𝑖=1,...,𝑛, (18) problem of this paper as a summer undergraduate research ∏𝑛 (𝜆 −𝜆 ) project, “hypothesize, and then verify, if possible, the explicit 1≤𝑗≤𝑛, 𝑗 =𝑖̸ 𝑖 𝑗 entries of an 𝑛×𝑛symmetric, tridiagonal matrix with −1/2,−1/2 eigenvalues (15), such that the eigenvalues of its principal determine the Hahn polynomials, ℎ𝑘 (𝑥/2, 𝑛), 𝑘= submatrix are (16).” Meanwhile the misplaced fax from the 0,1,...,𝑛 −, 1 whose three-term recurrence coefficients are first author’s proof was found during an office cleaning. The the entries of a Jacobi matrix with eigenvalues (3), hence student, A. De Serre-Rothney, was able to complete both parts similar to our 𝐴(𝑛). 4 Journal of Applied Mathematics

as 𝑛→∞. To demonstrate this, we will explicitly solve for k k k k k 1 2 3 4 n the stiffnesses and masses associated with 𝐵(𝑛). ··· With 𝐵(𝑛) = 𝐴(𝑛) +𝐼 we note that m1 m2 m3 mn 𝐵 =𝑎 =𝑛, 𝑖=1,...,𝑛 (a) 𝑖𝑖 𝑖 1 k1 k2 k3 k4 kn 𝐵 =−𝑏 =− √𝑖 (2𝑛−𝑖−1), 𝑖=1,...𝑛−2 𝑖,𝑖+1 𝑖 2 (20) ··· 𝑛 (𝑛−1) m1 m2 m3 𝐵 =−𝑏 =−√ 𝑛−1,𝑛 𝑛−1 2 (b) 𝑛−1 𝑜 Figure 1: Spring-mass system: (a) right hand end free, (b) right hand with eigenvalues {2𝑘 + 1}𝑘=0, while 𝐵 (𝑛) has eigenvalues 𝑛−1 end fixed. {2𝑘}𝑘=1. 1/2 1/2 𝑇 Let u =⟨𝑚1 ,...,𝑚𝑛 ⟩ with 𝑚𝑖 >0for all 𝑖.Let𝑚= ∑𝑛 𝑚 = u𝑇u 4. A Spring-Mass Model Problem 𝑖=1 𝑖 .Wewishtosolve

Onesimpleproblemwheresymmetrictridiagonalmatrices −1/2 𝑇 arise naturally is the inverse problem for the spring-mass 𝐵 (𝑛) u =⟨𝑚1 𝑘𝑖,0,...,0⟩ (21) systemshowninFigure1. In this case the squares of the natural frequencies of free vibration for system (a) are the (𝑚 )𝑛 𝑘 eigenvalues of a Jacobi matrix 𝐵, while those for system (b) for 𝑖 𝑖=1 and 1. 𝑜 𝑛 are the eigenvalues of its principal minor 𝐵 . The bottom, th, equation is Specifically, let 𝐶 be the stiffness matrix, and let 𝑀 be the mass (inertia) matrix for the system in Figure 1(a): −𝑛𝑚1/2 𝑛 1/2 𝑚1/2 = 𝑛 = √2( ) 𝛼, (22) 𝑛−1 −𝑏 𝑛−1 𝑘1 +𝑘2 −𝑘2 𝑛−1 [ ] [ −𝑘2 𝑘2 +𝑘3 −𝑘3 ] 𝐶=[ ⋅⋅ ⋅ ] , [ ] 𝑚1/2 =𝛼 [ −𝑘 𝑘 +𝑘 −𝑘 ] wherewechoose 𝑛 .Wewillthusbeabletoexpress 𝑛−1 𝑛−1 𝑛 𝑛 𝑚1/2 𝛼 [ −𝑘𝑛 𝑘𝑛 ] 𝑖 in terms of the scaling parameter . (19) The (𝑛 − 1)th equation is 𝑚1 [ ] [ 𝑚2 ] 1/2 [ ] 𝛼𝑏 −𝑛𝑚 √𝑛 (𝑛−1) /2 − 𝑛√2√𝑛/ (𝑛−1) 𝑀=[ ⋅ ] . 𝑚1/2 = 𝑛−1 𝑛 =𝛼 [ ] 𝑛−2 𝑚𝑛−1 −𝑏𝑛−2 − (1/2) √(𝑛−2)(𝑛+1) [ 𝑚𝑛] 𝑛(𝑛 + 1) 1/2 =𝛼√2( ) . Then the squares of the natural frequencies of the systems (𝑛 − 1)(𝑛 − 2) 𝑜 𝑜 𝑜 𝑜 in Figure 1 satisfy (𝐶 − 𝜆𝑀)x = 0 and (𝐶 −𝜆𝑀 )x = 0, 𝑜 (23) where 𝐶 is obtained from 𝐶 by deleting the last row and 0 column. The solutions can be ordered 0<𝜆1 <𝜆1 < 𝑜 𝑖 𝑖 =1̸ 𝑛−1 𝑛 𝜆2 < ⋅⋅⋅ < 𝜆𝑛−1 <𝜆𝑛−1 <𝜆𝑛.Wecanalsorewritethe The th equation, for , , ,is 𝑜 𝑜 𝑜 systems as (𝐵 − 𝜆𝐼)u = 0 and (𝐵 −𝜆𝐼)u = 0 where −1/2 −1/2 1/2 𝐵=𝑀 𝐶𝑀 and u =𝑀 x.Notethatthesquaresof 1/2 1/2 1/2 −𝑏𝑖−1𝑚𝑚−𝑖 +𝑛𝑚𝑖 −𝑏𝑖𝑚𝑖+1 =0. (24) the natural frequencies of the systems are the eigenvalues of 𝑜 𝐵 and 𝐵 . Suppose that the matrix 𝐵(𝑛) := 𝐴(𝑛) +𝐼 was to Then arise from a spring-mass system like in Figure 1;thatis, we are considering the system whose squares of the natural 2𝑛𝑚1/2 −(𝑖(2𝑛−𝑖−1))1/2𝑚1/2 {1,3,...,2𝑛− 1} 𝑚1/2 = 𝑖 𝑖+1 . frequencies are the equally spaced values 𝑖−1 1/2 (25) for system (a) and {2,4,...,2𝑛 − 2} for system (b). The ((𝑖 − 1)(2𝑛 − 𝑖)) system in Figure 1 is the simplest possible discrete model for a rod vibrating in longitudinal motion and more closely Now suppose approximates the continuous system as 𝑛→∞.Inaphysical system, we expect clustering of frequencies. The test matrix 1/2 𝐵(𝑛) does not share this phenomenon and so we expect the 1/2 𝑛 (𝑛+1) ⋅⋅⋅(𝑛+𝑖−1) 𝑚 =𝛼√2( ) (26) stiffnesses and masses associated with it to become unrealistic 𝑛−𝑖 (𝑛−1)(𝑛−2) ⋅⋅⋅(𝑛−𝑖) Journal of Applied Mathematics 5

1/2 1/2 for 𝑖=1,2,...,𝑗.Thencases𝑖=1,2are already verified, and Since 𝐶=𝑀 𝐵(𝑛)𝑀 ,then the strong inductive assumption applied in (25)with𝑖−1= 𝑛 − (𝑗 +1) 𝑖=𝑛−𝑗 1/2 1/2 implies .So 𝑘𝑖+1 =−𝐶𝑖,𝑖+1 =−𝑚𝑛−(𝑛−𝑖)𝐵𝑖,𝑖+1𝑚𝑛−(𝑛−𝑖−1) 𝑖! (2𝑛−𝑖−1)! (32) 𝑛(𝑛+1)⋅⋅⋅(𝑛+𝑗−1) 1/2 =𝛼2 ( ), 𝑚 =(2𝑛𝛼√2( ) ((𝑛−1)!)2 𝑛−𝑗−1 (𝑛−1)(𝑛−2)⋅⋅⋅(𝑛−𝑗) (2𝑛 − 1)! 𝑘 =𝛼2 . 1/2 1/2 1 ((𝑛 − 1)!)2 (33) −((𝑛 − 𝑗)(𝑛 + 𝑗 −1)) 𝑚𝑛−𝑗+1)

2 −1 From (26)wehave𝑚1/𝑚𝑛 = 2((2𝑛−2)!/((𝑛 − 1)!) ) which × (((𝑛 − 𝑗 − 1)(𝑛 +𝑗))1/2) goes to infinity as 𝑛→∞and from (32)weseethat𝑘1/𝑘𝑛 = (2𝑛 − 1)!/(𝑛 − 1)!𝑛! whichalsogoestoinfinityas𝑛→∞. 𝑛 (𝑛+1) ⋅⋅⋅(𝑛+𝑗−1) 1/2 =𝛼√2( ) Thisisnotamodelofaphysicalrod,asexpected. (𝑛−1)(𝑛−2) ⋅⋅⋅(𝑛−𝑗)

𝑛+𝑗−1 1/2 5. Conclusion ×[(2𝑛( ) 𝑛−𝑗 Afamilyof𝑛×𝑛symmetric tridiagonal matrices, 𝑊𝑛,whose eigenvalues are simple and uniformly spaced and whose −((𝑛 − 𝑗)(𝑛 + 𝑗 −1))1/2) leading principle submatrix has uniformly interlaced, simple eigenvalues has been presented (14). Members of the family are characterized by a specified smallest eigenvalue 𝑎𝑜 and gap −1 𝑐 ×(((𝑛 − 𝑗 − 1)(𝑛1/2 +𝑗)) ) ] size between eigenvalues. The matrices are termed Jacobian, since the off-diagonal entries are all nonzero. The matrix entries are explicit functions of the size 𝑛, 𝑎𝑜,and𝑐;sothe 𝑛 (𝑛+1) ⋅⋅⋅(𝑛+𝑗−1) 1/2 matrices can be used as a test matrices for eigenproblems, =𝛼√2( ) (𝑛 − 1)(𝑛 − 2) ⋅ ⋅ ⋅ (𝑛 −𝑗) both forward and inverse. The matrix 𝑊𝑛 for specified smallest eigenvalue 𝑎𝑜 and gap 𝑐 is unique up to the signs of 2𝑛 − (𝑛 − 𝑗) the off-diagonal elements. ×[ ] 𝑊 ((𝑛 − 𝑗 − 1)(𝑛 +𝑗))1/2 In Section 4,theformof 𝑛 wasusedasanexplicit solution of a spring-mass vibration model (Figure 1), and 𝑛 (𝑛+1) ⋅⋅⋅(𝑛+𝑗−1)(𝑛+𝑗) 1/2 the inverse problem to determine the lumped masses and =𝛼√2( ) spring stiffnesses was solved explicitly. Both the lumped (𝑛−1)(𝑛−2) ⋅⋅⋅(𝑛−𝑗)(𝑛−𝑗−1) masses 𝑚𝑛−𝑖 given by (30) and spring stiffnesses 𝑘𝑛−𝑖 from (32) (27) show superexponential growth. Consequently 𝑚𝑛/𝑚1, 𝑘𝑛/𝑘1 become vanishingly small as 𝑛→∞. As a result, the spring- 1/2 mass system of Figure 1 cannot be used as a discretized model which verifies, by strong induction, the closed form for 𝑚𝑛−𝑖 for a physical rod in longitudinal vibration, as the model given by (26). 𝑛→∞ Finally, the first equation of21 ( )is becomes unrealistic in the limit as .

1/2 1/2 −1/2 Conflict of Interests 𝑛𝑚1 −𝑏1𝑚2 =𝑚1 𝑘1 (28) The authors declare that there is no conflict of interests and so regarding the publication of this paper.

1/2 𝑘1 =𝑛𝑚1 −𝑏1(𝑚1𝑚2) . (29) References

[1] H. Hochstadt, “On some inverse problems in matrix theory,” We note that the values 𝑚𝑛−𝑖 canbewrittenas Archiv der Mathematik,vol.18,pp.201–207,1967. [2]G.M.L.Gladwell,Inverse Problems in Vibration,vol.9of (𝑛+𝑖−1)! (𝑛−𝑖−1)! 𝑚 =2𝛼2 Monographs and Textbooks on Mechanics of Solids and Fluids: 𝑛−𝑖 ((𝑛−1)!)2 (30) Mechanics. Dynamical Systems, Martinus Nijhoff Publishers, Dordrecht, The Netherlands, 1986. 𝑖=1,...,𝑛−1 [3] N. Brad Willms, Some matrix inverse eigenvalue problems [M.S. for ,and thesis], University of Waterloo, Ontario, Canada, 1988. [4] F. W. Biegler-Konig,¨ “Construction of band matrices from 2 (𝑛+0−1)! (𝑛−0−1)! 2 spectral data,” Linear Algebra and Its Applications,vol.40,pp. 𝑚𝑛 =𝛼 =𝛼. (31) ((𝑛 − 1)!)2 79–87, 1981. 6 Journal of Applied Mathematics

[5] C. de Boor and G. H. Golub, “The numerically stable recon- struction of a Jacobi matrix from spectral data,” Linear Algebra and Its Applications,vol.21,no.3,pp.245–260,1978. [6] G.M.L.GladwellandN.B.Willms,“AdiscreteGel’fand-Levitan method for band-matrix inverse eigenvalue problems,” Inverse Problems,vol.5,no.2,pp.165–179,1989. [7]D.BoleyandG.H.Golub,“Inverseeigenvalueproblemsfor band matrices,” in NumericalAnalysis(Proc.7thBiennialConf., Univ. Dundee, Dundee, 1977),G.A.Watson,Ed.,vol.630of Lecture Notes in Math., pp. 23–31, Springer, Berlin, Germany, 1978. [8] O. H. Hald, “Inverse eigenvalue problems for Jacobi matrices,” Linear Algebra and Its Applications,vol.14,no.1,pp.63–85,1976. [9] H. Hochstadt, “On the construction of a Jacobi matrix from spectral data,” Linear Algebra and Its Applications,vol.8,pp. 435–446, 1974. [10] G. H. Golub and C. F. Van Loan, Matrix Computations,Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, Md, USA, 4th edition, 2013. [11] S.-G. Hwang, “Cauchy’s interlace theorem for eigenvalues of Hermitian matrices,” The American Mathematical Monthly,vol. 111, no. 2, pp. 157–159, 2004. [12] S. Fisk, “A very short proof of Cauchy’s interlace theorem for eigenvalues of Hermitian matrices,” The American Mathemati- cal Monthly,vol.112,no.2,p.118,2005. [13] A. De Serre Rothney, “Eigenvalues of a special tridiagonal matrix,” 2013, http://www.ubishops.ca/fileadmin/bishops doc- uments/natural sciences/mathematics/files/2013-De Serre.pdf. [14] P. A. Clement, “A class of triple-diagonal matrices for test purposes,” SIAM Review,vol.1,pp.50–52,1959. [15] A.Edelman,E.Kostlan,and.In,“TheroadfromKac’smatrixto Kac’s random polynomials,” in Proceedings of the SIAM Applied Linear Algebra Conference, pp. 503–507, Philadelphia, Pa, USA, 1994. [16] T. Muir, ATreatiseontheTheoryofDeterminants,Dover,New York, NY, USA, 196 0. [17] O. Taussky and J. Todd, “Another look at a matrix of Mark Kac,” Linear Algebra and Its Applications,vol.150,pp.341–360. [18] A. F. Nikiforov, S. K. Suslov, and V. B. Uvarov, Classical Orthogonal Polynomials of a Discrete Variable, Springer Series in Computational Physics, Springer, Berlin, Germany, 1991. Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2014, Article ID 601847, 9 pages http://dx.doi.org/10.1155/2014/601847

Research Article Natural Filtrations of Infinite-Dimensional Modular Contact Superalgebras

Qiang Mu

School of Mathematical Sciences, Harbin Normal University, Harbin 150025, China

Correspondence should be addressed to Qiang Mu; [email protected]

Received 19 December 2013; Accepted 24 March 2014; Published 24 April 2014

Academic Editor: P. N. Shivakumar

Copyright © 2014 Qiang Mu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The natural filtration of the infinite-dimensional contact superalgebra over an algebraic closed field of positive characteristic is proved to be invariant under automorphisms by characterizing ad-nilpotent elements and the subalgebras generated by certain ad- nilpotent elements. Moreover, we obtain an intrinsic characterization of contact superalgebras and a property of automorphisms of these Lie superalgebras.

1. Introduction Theorem 1. The natural filtration of the infinite-dimensional contact Lie superalgebra is invariant under automorphisms. Filtration structures play an important role in the classification of modular Lie algebras (see [1, 2]) and nonmodular Thereby, one obtains the following theorems. Lie superalgebras (see [3, 4]), respectively. We know that the 󸀠 󸀠 Theorem 2. Suppose that 𝑟, 𝑟 , 𝑛, 𝑛 are positive integers. Then Lie algebras and Lie superalgebras of Cartan type possess a 󸀠 󸀠 󸀠 󸀠 natural filtration structure. The natural filtrations of finite- 𝐾(2𝑟 + 1, 𝑛) ≅ 𝐾(2𝑟 +1,𝑛) if and only if (𝑟, 𝑛) = (𝑟 ,𝑛 ). dimensionalmodularLiealgebrasofCartantypewereproved Theorem 3. 𝜙 𝜓 𝐾(2𝑟 + 1, 𝑛) to be invariant in [5, 6]. In the infinite-dimensional case, Let , be automorphisms of .Then 𝜙=𝜓ifandonlyif𝜙|𝐾 =𝜓|𝐾 . thesameconclusionwasprovedin[7], by determining ad- [−1] [−1] nilpotent elements. In the case of Lie superalgebras of Cartan The paper is organized as follows. In Section 2,werecall type, the invariance of the natural filtrations of some Lie the necessary definitions concerning the modular contact superalgebras was proved in [8, 9]. Similar results for Lie superalgebra 𝐾(2𝑟 + 1, 𝑛).InSection3,westudythead- superalgebras of generalized Cartan type were obtained in nilpotent elements of 𝐾(2𝑟 + 1, 𝑛).InSection4,wecomplete [10–12], respectively. the proofs of Theorems 1–3. In this paper, we consider the infinite-dimensional modular contact superalgebra 𝐾(2𝑟 + 1, 𝑛),whichisanalogous to the one in the nonmodular situation (see [13]). But since 2. Preliminaries the principal Z-gradations of Lie superalgebras of Cartan type are different (see [13]), most results and proofs for other Throughout this paper, F denotes an algebraic closed field of Lie superalgebras cannot be applied to contact superalgebras. characteristic 𝑝>2and Z2 ={0, 1} the ring of integers Therefore the corresponding results and proofs for contact modulo 2. Let N and N0 denote the sets of positive integers superalgebras have to be established separately. By determin- and nonnegative integers, respectively. Given 𝑚∈N, 𝑚>1 𝑚 𝑚 ing the ad-nilpotent elements and subalgebras generated by and 𝛼=(𝛼1,𝛼2,...,𝛼𝑚)∈N0 ,weput|𝛼| = ∑𝑖=1 𝛼𝑖.Let certain ad-nilpotent elements, we prove the main result of this O(𝑚) denote the divided power algebra over F with basis (𝛼) 𝑚 paper. {𝑥 |𝛼∈N0 }.For𝜀𝑖 =(𝛿𝑖1,𝛿𝑖2,...,𝛿𝑖𝑚), 𝑖 = 1,2,...,𝑚,we 2 Journal of Applied Mathematics

(𝜀𝑖) abbreviate 𝑥 to 𝑥𝑖.LetΛ(𝑛) be the exterior superalgebra Hereafter let 𝑟 be a positive integer and let 𝑚=2𝑟+1.Put over F in 𝑛 variables 𝑥𝑚+1,𝑥𝑚+2,...,𝑥𝑠,where𝑠=𝑚+𝑛. 𝐽=𝑌\{𝑚}and 𝐽0 =𝑌0 \{𝑚}.For𝑖∈𝐽, define Denote by O(𝑚, 𝑛) the tensor product O(𝑚)⊗Λ(𝑛).Thetrivial Z O(𝑚) Z Λ(𝑛) 𝑖+𝑟 1≤𝑖≤𝑟 11≤𝑖≤𝑟 2-gradation of and the natural 2-gradation of { { Z O(𝑚, 𝑛) O(𝑚, 𝑛) 󸀠 induce a 2-gradation of such that is an 𝑖 = {𝑖−𝑟 𝑟<𝑖≤2𝑟 𝜎 (𝑖) = {−1 𝑟<𝑖≤2𝑟 (8) associative superalgebra. For 𝑔∈O(𝑚) and 𝑓∈Λ(𝑛),we { { 𝑚 {𝑖𝑚<𝑖≤𝑠, {1𝑖∈𝑌1. abbreviate 𝑔⊗𝑓to 𝑔𝑓.For𝛼, 𝛽 ∈ N0 and 𝑖, 𝑗 = 𝑚 + 1, 𝑚+ 2,...,𝑠,thefollowingformulasinO(𝑚, 𝑛) hold: Let 𝐷𝐾 : O(𝑚, 𝑛) → 𝑊(𝑚, 𝑛) be the linear mapping such 𝛼+𝛽 that 𝑥(𝛼)𝑥(𝛽) =( )𝑥(𝛼+𝛽),𝑥𝑥 =−𝑥𝑥 , 𝛼 𝑖 𝑗 𝑗 𝑖 𝑠 (1) 𝐷𝐾 (𝑓) = ∑𝑓𝑖𝐷𝑖, (𝛼) (𝛼) (9) 𝑥 𝑥𝑗 =𝑥𝑗𝑥 , 𝑖=1 where where 𝑚 𝜇(𝑖) deg(𝑓) 󸀠 𝛼+𝛽 𝛼 +𝛽 𝑓 = (−1) (𝑥 𝐷 (𝑓) + 𝜎 (𝑖 )𝐷󸀠 (𝑓)) , ∀𝑖 ∈ 𝐽, ( )=∏ ( 𝑖 𝑖). 𝑖 𝑖 𝑚 𝑖 𝛼 𝛼 (2) 𝑖=1 𝑖 𝑓𝑚 =2𝑓−∑𝑥𝑖𝐷𝑖 (𝑓) . 𝑖∈𝐽 Put 𝑌0 = {1,2,...,𝑚}, 𝑌1 = {𝑚+ 1,𝑚 + 2,...,𝑠},and 𝑌=𝑌0 ∪𝑌1.Let (10)

B𝑘 ={(𝑖1,𝑖2,...,𝑖𝑘)|𝑚+1≤𝑖1 <𝑖2 <⋅⋅⋅<𝑖𝑘 ≤𝑠} , Then

𝑛 (3) [𝐷𝐾 (𝑓) ,𝐷𝐾 (𝑔)] =𝐷𝐾 (⟦𝑓, 𝑔⟧) , (11) B (𝑛) = ⋃B𝑘, 𝑘=0 where ⟦𝑓,𝑔⟧𝐾 =𝐷 (𝑓)(𝑔) 𝑚− 2𝐷 (𝑓)𝑔 (see [14]). It follows directly from (11)andtheinjectivityof𝐷𝐾 that ⟦, ⟧ defines a where B0 =⌀.Given𝑢=(𝑖1,𝑖2,...,𝑖𝑘)∈B𝑘,set|𝑢| =, 𝑘 {𝑢} = {𝑖 ,𝑖 ,...,𝑖 } 𝑥𝑢 =𝑥 𝑥 ⋅⋅⋅𝑥 (|⌀| = 0, 𝑥⌀ =1) Lie multiplication on O(𝑚, 𝑛). This Lie superalgebra, denoted 1 2 𝑘 ,and 𝑖1 𝑖2 𝑖𝑘 . (𝛼) 𝑢 𝑚 by 𝐾(2𝑟 + 1, 𝑛), is called the infinite-dimensional contact Then {𝑥 𝑥 |𝛼∈N0 ,𝑢∈B(𝑛)} is an F-basis of the infinite- 𝐾 𝐾(2𝑟+1, 𝑛) O(𝑚, 𝑛) superalgebra. In the sequel, we simply write for . dimensional superalgebra . The following formula holds in 𝐾 (see [14]): Let 𝐷1,𝐷2,...,𝐷𝑠 be the linear transformations of O(𝑚, 𝑛) such that 𝜇(𝑖) deg(𝑓) ⟦𝑓,𝑔⟧ = ∑𝜎 (𝑖)(−1) 𝐷𝑖 (𝑓)𝑖 𝐷 󸀠 (𝑔) (𝛼−𝜀 ) 𝑢 𝑖∈𝐽 𝑥 𝑖 𝑥 𝑖∈𝑌 { 0 (𝛼) 𝑢 𝑢 𝐷𝑖 (𝑥 𝑥 )= 𝜕𝑥 { (𝛼) (4) +(2𝑓−∑𝑥 𝐷 (𝑓)) 𝐷 (𝑔) 𝑥 ⋅ 𝑖∈𝑌1. 𝑖 𝑖 𝑚 { 𝜕𝑥𝑖 𝑖∈𝐽 𝐷 ,𝐷 ,...,𝐷 Then 1 2 𝑠 are superderivations of the superalgebra deg(𝑓) deg(𝑔) O(𝑚, 𝑛).Let − (−1) (2𝑔 − ∑𝑥𝑖𝐷𝑖 (𝑔))𝑚 𝐷 (𝑓) . 𝑖∈𝐽 𝑠 (12) 𝑊 (𝑚, 𝑛) ={∑𝑎𝑖𝐷𝑖 |𝑎𝑖 ∈ O (𝑚, 𝑛) ,𝑖∈𝑌}. (5) 𝑖=1 ∞ Then 𝐾=⨁𝑖=−2𝐾[𝑖] is a Z-graded Lie superalgebra, where 𝑊(𝑚, 𝑛) Then is an infinite-dimensional Lie superalgebra (𝛼) 𝑢 contained in Der(O(𝑚, 𝑛)) (see [14]). If deg(𝑥) appears in 𝐾[𝑖] = spanF {𝑥 𝑥 | |𝛼| +𝛼𝑚 + |𝑢| =𝑖+2} . (13) some expression in this paper, we always regard 𝑥 as a Z2- 𝐾 = ⨁ 𝐾 𝑗≥−2 𝐾=𝐾 ⊃𝐾 ⊃𝐾 ⊃ homogeneous element and deg(𝑥) as the Z2-degree of 𝑥. Let 𝑗 𝑖≥𝑗 [𝑖] for .Then −2 −1 0 Then deg(𝐷𝑖)=𝜇(𝑖),where ⋅⋅⋅ is referred to as the natural filtration of 𝐾. ∞ 0𝑖∈𝑌0 Lemma 4. 𝐾=⨁𝑖=−2𝐾[𝑖] is transitively graded. 𝜇 (𝑖) ={ (6) 1𝑖∈𝑌1. Proof. Assume the contrary, then there exists 𝑦∈𝐾[ℓ] such ⟦𝑦, 𝑥 ⟧=0 𝑗∈𝐽 ℓ≥0 The following formula holds in 𝑊(𝑚, 𝑛) (see [14]): that 𝑗 for all ,where . Suppose that the largest exponent of 𝑥𝑚 among the nonzero summands in the deg(𝑎𝐷𝑖) deg(𝑏𝐷𝑗) 𝑦 𝑡 [𝑎𝐷𝑖,𝑏𝐷𝑗]=𝑎𝐷𝑖 (𝑏) 𝐷𝑗 − (−1) 𝑏𝐷𝑗 (𝑎) 𝐷𝑖, expression of is equal to ,andwrite (7) 𝑦= ∑ 𝑐 𝑥(𝛼)𝑥𝑢 + ∑ 𝑑 𝑥(𝛽)𝑥V, 𝛼,𝑢 𝛽,V (14) 𝛼,𝑢,𝛼 =𝑡 𝛽,V,𝛽 <𝑡 where 𝑎, 𝑏 ∈ O(𝑚, 𝑛) and 𝑖, 𝑗 ∈𝑌. 𝑚 𝑚 Journal of Applied Mathematics 3

where 𝑐𝛼,𝑢,𝑑𝛽,V ∈ F.Hence,for𝑗∈𝐽0, (2) Suppose that 𝑦[−2] =0̸.As𝑦 is ad-nilpotent, 𝑦[−2] is ad- nilpotent by (1).Notethat 󸀠 (𝛼−𝜀 ) 𝑢 0=⟦𝑦,𝑥⟧= ∑ 𝜎(𝑗 )𝑐 𝑥 𝑗 𝑥 +ℎ, 𝑗 𝛼,𝑢 𝑘 (𝑘𝜀 ) 𝑘−1 ((𝑘−1)𝜀 ) 𝑘 (15) ( 1) (𝑥 𝑚 ) = ( 1) (2𝑥 𝑚 ) =2 =0,̸ 𝛼,𝑢,𝛼𝑚=𝑡 ad ad (19) where each exponent of 𝑥𝑚 of all nonzero summands for all 𝑘>0.Thisshowsthat𝑦[−2] is not ad-nilpotent, a in the expression of ℎ is less than 𝑡.Then contradiction. 󸀠 (𝛼−𝜀𝑗) 𝑢 (3) (1) 𝑦 ∑𝛼,𝑢,𝛼 =𝑡 𝜎(𝑗 )𝑐𝛼,𝑢𝑥 𝑥 =0.Notethatallnonzero By ,weseethat [−1] is ad-nilpotent. Suppose that 𝑚 𝑦 =∑ 𝑎 𝑥 =0̸ 𝑎 ∈ F 󸀠 (𝛼−𝜀𝑗) 𝑢 [−1] 𝑖∈𝐽 𝑖 𝑖 ,where 𝑖 . Then there exists some ∑ 𝜎(𝑗 )𝑐𝛼,𝑢𝑥 𝑥 F 0 summands of 𝛼,𝑢,𝛼𝑚=𝑡 are -linear 𝑎 =0̸ (𝛼−𝜀 ) 𝑢 𝑗 . A direct calculation shows that independent. It follows that each 𝑥 𝑗 𝑥 is equal to 0. (𝛼) 𝑢 𝑥 𝑥 𝑥 0 𝑘 (𝑘𝜀 󸀠 ) 𝑘−1 ((𝑘−1)𝜀 󸀠 ) Hence the exponents of 𝑗 in each are equal to . ( 𝑦 ) (𝑥 𝑗 )=( 𝑦 ) (𝜎 (𝑗) 𝑎 𝑥 𝑗 ) 𝑗∈𝑌 𝑥 ad [−1] ad [−1] 𝑗 Similarly, for 1,wecanprovethat 𝑗 does not appear (20) 𝑥(𝛼)𝑥𝑢 𝑥(𝛼)𝑥𝑢 𝑘 𝑘 in each . Consequently, we see that all are of = 𝜎(𝑗) 𝑎𝑗 =0,̸ (𝑡𝜀𝑚) the form 𝑥 .If𝑡=0,then𝑦∈𝐾[−2], contradicting ℓ≥0. Hence 𝑡>0,andwecanwrite for all 𝑘>0.Itfollowsthat𝑦[−1] is not ad-nilpotent. (4) is an immediate consequence of (2), (3),and(1). (5) (𝑡𝜀 ) (𝛼) 𝑢 (𝛽) V 𝑦=𝑐 𝑥 𝑚 + ∑ 𝑐 𝑥 𝑥 + ∑ 𝑑 𝑥 𝑥 , (2) (1) (3) 𝑡𝜀𝑚 𝛼,𝑢 𝛽,V follows from , , and the proof of . 𝛼,𝑢,𝛼𝑚=𝑡−1 𝛽,V,𝛽𝑚<𝑡−1 𝑎∈N 𝑎=∑∞ 𝑎 𝑝𝑙 𝑝 (16) Let 0 and 𝑙=0 𝑙 be the -adic expression of 𝑎,where0≤𝑎𝑙 <𝑝.Then, 𝑐 =0̸ 𝑗∈𝑌 where 𝑡𝜀𝑚 .Notethat,for 1, pad (𝑎) =(pad0 (𝑎) , pad1 (𝑎) , pad2 (𝑎) ,...) (21)

((𝑡−1)𝜀𝑚) 0=⟦𝑦,𝑥𝑗⟧=−𝑐𝑡𝜀 𝑥 𝑥𝑗 𝑚 is said to be the 𝑝-adic sequence of 𝑎,wherepad𝑗(𝑎) =𝑗 𝑎 for 𝑗∈N 𝛼=(𝛼,𝛼 ,...,𝛼 )∈N𝑚 𝑝 + ∑ (−1)|𝑢|𝑐 𝑥(𝛼)𝐷 (𝑥𝑢)+ℎ, (17) all 0.For 1 2 𝑚 0 , define the -adic 𝛼,𝑢 𝑗 matrix of 𝛼 to be 𝛼,𝑢,𝛼𝑚=𝑡−1

pad (𝛼1) where each exponent of 𝑥𝑚 of the nonzero summands in the (𝛼 ) expression of ℎ is less than 𝑡−1. Therefore, (𝛼) =(pad 2 ). pad . (22) . ((𝑡−1)𝜀𝑚) |𝑢| (𝛼) 𝑢 −𝑐𝑡𝜀 𝑥 𝑥𝑗 + ∑ (−1) 𝑐𝛼,𝑢𝑥 𝐷𝑗 (𝑥 )=0. 𝑚 (18) pad (𝛼𝑚) 𝛼,𝑢,𝛼𝑚=𝑡−1 As pad(𝛼) is an 𝑚×∞matrix with finitely many nonzero ((𝑡−1)𝜀𝑚) Since 𝑥𝑗 appears in 𝑥 𝑥𝑗 and does not appear in elements, |𝑢| (𝛼) 𝑢 ∑𝛼,𝑢,𝛼 =𝑡−1 (−1) 𝑐𝛼,𝑢𝑥 𝐷𝑗(𝑥 ),weconcludethat𝑡=0,a 𝑚 (𝛼) = {𝑗 ∈ N |∃𝑖∈𝑌 : (𝛼 ) =0}̸ contradiction. ht max 0 0 pad𝑗 𝑖 (23) is well defined. Let 3. ad-Nilpotent Elements 𝑚 𝑐 ‖𝛼‖ = ∑∑ (𝛼 ), Recall that 𝑦∈𝐾is called ad-nilpotent if there exists 𝑡∈N 𝑏,𝑐 pad𝑗 𝑖 (24) 𝑡 𝑖=1 such that (ad 𝑦) (𝐾) =.Forasubset 0 𝑅 of 𝐾,letnil(𝑅) denote 𝑗=𝑏 𝑅 (𝑅) the set of ad-nilpotent elements in ,andletNil denote the 𝑏∈N 𝑐∈N ‖𝛼‖ ‖𝛼‖ 𝐾 (𝑅) for 0 and .Weabbreviate 0,𝑞 to 𝑞. subalgebra of generated by nil . (𝛼) 𝑢 Suppose that 𝑦=∑𝛼,𝑢 𝑐𝛼,𝑢𝑥 𝑥 is a nonzero element of 𝐾 𝑐 ∈ F Lemma 5. Suppose that 𝑦[𝑖] ∈𝐾[𝑖] for 𝑖≥−2.Thefollowing ,where 𝛼,𝑢 .Define statements hold. (𝑦) = { (𝛼) |𝑐 =0}.̸ 𝑡 ht max ht 𝛼,𝑢 (25) (1) If 𝑦=∑𝑖=𝑘 𝑦[𝑖] ∈ nil(𝐾),then𝑦[𝑘] ∈ nil(𝐾). (𝛼) 𝑢 𝑡 Given that 𝑞>0and 𝑥 𝑥 ∈𝐾, we define (2) If 𝑦=∑𝑖=−2 𝑦[𝑖] ∈ nil(𝐾),then𝑦[−2] =0. 𝑡 F (𝑥(𝛼)𝑥𝑢) = ‖𝛼‖ +2‖𝛼‖ + |𝑢| + (𝛼 ) . (3) If 𝑦=∑𝑖=−2 𝑦[𝑖] ∈ nil(𝐾0),then𝑦[−1] =0. 𝑞 𝑞 1,𝑞 pad0 𝑚 (26) 𝑡 (4) If 𝑦=∑ 𝑦[𝑖] ∈ nil(𝐾 ),then𝑦[0] ∈ nil(𝐾 ). 𝑚 𝑖=−2 0 0 Lemma 6. Let 𝛼, 𝛽 ∈ N0 , 𝑖∈𝑌0,and𝑞∈N.Then, 𝑡 (5) If 𝑦=∑ 𝑦[𝑖] ∈ nil(𝐾),then𝑦[−1] ∈ span {𝑥𝑗 |𝑗∈ (𝛼) (𝛽) 𝑖=−2 F (1) 𝑥 𝑥 =0̸if and only if pad(𝛼)+pad(𝛽) = pad(𝛼+𝛽); 𝑌1}. (2) If 𝛽𝑖 =0̸,then‖𝛽 −𝑖 𝜀 ‖𝑞 +2‖𝛽−𝑖 𝜀 ‖1,𝑞 ≥‖𝛽‖𝑞 +2‖𝛽‖1,𝑞 − Proof. (1) See [15,Lemma5.1]. 1; 4 Journal of Applied Mathematics

(𝛽) V (𝛽) V (𝛽) V (𝛼) 𝑢 (𝛽) V (𝛼) 𝑢 (𝛽) V (3) If 𝑥 𝑥 ∈𝐾1 and 𝑞≥ht(𝑥 𝑥 ),thenF𝑞(𝑥 𝑥 )≥ (2) If 𝑥 𝑥 𝐷𝑚(𝑥 𝑥 ) =0̸,thenF𝑞(𝑥 𝑥 𝐷𝑚(𝑥 𝑥 )) ≥ 3 (𝛼) 𝑢 . F𝑞(𝑥 𝑥 )+1. (1) (𝛽) V (𝛼) 𝑢 (𝛽) V Proof. See [7,Lemma2.5]. (3) If 𝐷𝑖 (𝑥 𝑥 )𝐷𝑖󸀠 (𝑥 𝑥 ) =0̸,thenF𝑞(𝐷𝑖(𝑥 𝑥 ) (2) First consider the case pad0(𝛽𝑖) =0̸.Then, (𝛼) 𝑢 (𝛼) 𝑢 𝐷𝑖󸀠 (𝑥 𝑥 )) ≥ F𝑞(𝑥 𝑥 )+1. (𝛽𝑖 −1) = ( (𝛽𝑖) −1, (𝛽𝑖) ,...) . pad pad0 pad1 (27) (𝛽) V (𝛼) 𝑢 Proof. (1) The assumption 𝑥 𝑥 𝐷𝑚(𝑥 𝑥 ) =0̸implies that ‖𝛽 − 𝜀 ‖ =‖𝛽‖ −1 ‖𝛽 − 𝜀 ‖ =‖𝛽‖ (𝛽) (𝛼−𝜀𝑚) It follows that 𝑖 𝑞 𝑞 and 𝑖 1,𝑞 1,𝑞, 𝑥 𝑥 =0̸.ByLemma6(1),wehave and thus (2) holds. (𝛽 )=0 Next consider the case pad0 𝑖 . We may assume that pad (𝛽+(𝛼−𝜀𝑚)) = pad (𝛽) + pad (𝛼−𝜀𝑚) . (33)

pad (𝛽𝑖) (28) Consequently, =(0,...,0,pad𝑡 (𝛽𝑖),pad𝑡+1 (𝛽𝑖),...), pad0 (𝛽𝑚 +(𝛼𝑚 −1))=pad0 (𝛽𝑚)+pad0 (𝛼𝑚 −1), (34) where pad𝑡(𝛽𝑖) =0̸and 𝑡≥1.Hence, 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩𝛽+(𝛼−𝜀 )󵄩 = 󵄩𝛽󵄩 + 󵄩𝛼−𝜀 󵄩 , 󵄩 𝑚 󵄩𝑞 󵄩 󵄩𝑞 󵄩 𝑚󵄩𝑞 (35) (𝛽 −1) pad 𝑖 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩𝛽+(𝛼−𝜀 )󵄩 = 󵄩𝛽󵄩 + 󵄩𝛼−𝜀 󵄩 . (29) 󵄩 𝑚 󵄩1,𝑞 󵄩 󵄩1,𝑞 󵄩 𝑚󵄩1,𝑞 (36) =(𝑝−1,...,𝑝−1,pad𝑡 (𝛽𝑖)−1,pad𝑡+1 (𝛽𝑖),...). By (2) and (3) of Lemma 6,weobtain If 𝑞<𝑡,then 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩𝛼−𝜀 󵄩 + 󵄩𝛼−𝜀 󵄩 ≥ ‖𝛼‖ + ‖𝛼‖ −1, 󵄩𝛽−𝜀𝑖󵄩𝑞 +2󵄩𝛽−𝜀𝑖󵄩1,𝑞 󵄩 𝑚󵄩𝑞 󵄩 𝑚󵄩1,𝑞 𝑞 1,𝑞 (37) (𝛽) V = (𝑞 + 1) (𝑝 − 1) + 2𝑞 (𝑝−1) F𝑞 (𝑥 𝑥 )≥3. (38) (30) >−1 Combining (34)–(38), we have 󵄩 󵄩 󵄩 󵄩 = 󵄩𝛽󵄩 +2󵄩𝛽󵄩 −1. 𝑞 1,𝑞 (𝛽) V (𝛼) 𝑢 F𝑞 (𝑥 𝑥 𝐷𝑚 (𝑥 𝑥 )) If 𝑞=𝑡,notingthat𝑝>2,then (𝛽+(𝛼−𝜀 )) V 𝑢 = F (𝑥 𝑚 𝑥 𝑥 ) 󵄩 󵄩 󵄩 󵄩 𝑞 󵄩𝛽−𝜀𝑖󵄩𝑞 +2󵄩𝛽−𝜀𝑖󵄩1,𝑞 󵄩 󵄩 󵄩 󵄩 = 󵄩𝛽+(𝛼−𝜀𝑚)󵄩𝑞 +2󵄩𝛽+(𝛼−𝜀𝑚)󵄩1,𝑞 + |𝑢| + |V| =𝑡(𝑝−1)+2(𝑡−1) (𝑝 − 1) + 3pad ( 𝑡 (𝛽𝑖)−1) + pad0 (𝛽𝑚 +(𝛼𝑚 −1)) ≥2+3pad𝑡 (𝛽𝑖)−3 (31) 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 = 󵄩𝛽󵄩𝑞 + 󵄩𝛼−𝜀𝑚󵄩𝑞 +2󵄩𝛽󵄩1,𝑞 +2󵄩𝛼−𝜀𝑚󵄩1,𝑞 + |𝑢| + |V| =3pad𝑡 (𝛽𝑖)−1 󵄩 󵄩 󵄩 󵄩 + pad0 (𝛽𝑚)+pad0 (𝛼𝑚 −1) = 󵄩𝛽󵄩𝑞 +2󵄩𝛽󵄩1,𝑞 −1. 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 = 󵄩𝛽󵄩𝑞 +2󵄩𝛽󵄩1,𝑞 + |V| + pad0 (𝛽𝑚)+󵄩𝛼−𝜀𝑚󵄩𝑞 If 𝑞>𝑡,then 󵄩 󵄩 +2󵄩𝛼−𝜀 󵄩 + |𝑢| + (𝛼 −1) 󵄩 󵄩 󵄩 󵄩 󵄩 𝑚󵄩1,𝑞 pad0 𝑚 󵄩𝛽−𝜀𝑖󵄩𝑞 +2󵄩𝛽−𝜀𝑖󵄩1,𝑞 (𝛽) V 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 ≥ F𝑞 (𝑥 𝑥 )+‖𝛼‖𝑞 +2‖𝛼‖1,𝑞 −1+|𝑢| + pad0 (𝛼𝑚)−1 = 󵄩𝛽−𝜀𝑖󵄩𝑡 +2󵄩𝛽−𝜀𝑖󵄩1,𝑡 +3󵄩𝛽−𝜀𝑖󵄩𝑡+1,𝑞 (32) (𝛼) 𝑢 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 ≥ F𝑞 (𝑥 𝑥 )+1. ≥ 󵄩𝛽󵄩𝑡 +2󵄩𝛽󵄩1,𝑡 −1+3󵄩𝛽−𝜀𝑖󵄩𝑡+1,𝑞 (39) 󵄩 󵄩 󵄩 󵄩 = 󵄩𝛽󵄩𝑞 +2󵄩𝛽󵄩1,𝑞 −1. (𝛼) 𝑢 (𝛽) V (2) Suppose that 𝑥 𝑥 𝐷𝑚(𝑥 𝑥 ) =0̸.Then, (𝛽) V (3) The assumption 𝑥 𝑥 ∈𝐾1 implies that |𝛽|+𝛽𝑚+|V|≥ 3. Then it is trivially verified that (3) holds. (𝛼) 𝑢 (𝛽) V F𝑞 (𝑥 𝑥 𝐷𝑚 (𝑥 𝑥 )) (𝛽) V (𝛼) 𝑢 Lemma 7. 𝑥 𝑥 ∈𝐾 𝑥 𝑥 ∈𝐾𝑞≥ (𝛼+(𝛽−𝜀 )) V 𝑢 Suppose that 1, , = F (𝑥 𝑚 𝑥 𝑥 ) (𝛽) V 𝑞 max{1, ht(𝑥 𝑥 )},and𝑖∈𝐽. The following statements hold. 󵄩 󵄩 󵄩 󵄩 (𝛽) V (𝛼) 𝑢 (𝛽) V (𝛼) 𝑢 = 󵄩𝛼+(𝛽−𝜀𝑚)󵄩𝑞 +2󵄩𝛼+(𝛽−𝜀𝑚)󵄩1,𝑞 + |𝑢| + |V| (1) If 𝑥 𝑥 𝐷𝑚(𝑥 𝑥 ) =0̸,thenF𝑞(𝑥 𝑥 𝐷𝑚(𝑥 𝑥 )) ≥ (𝛼) 𝑢 F𝑞(𝑥 𝑥 )+1. + pad0 (𝛼𝑚 +(𝛽𝑚 −1)) Journal of Applied Mathematics 5

󵄩 󵄩 󵄩 󵄩 (𝛽) V = ‖𝛼‖𝑞 + 󵄩𝛽−𝜀𝑚󵄩𝑞 +2‖𝛼‖1,𝑞 +2󵄩𝛽−𝜀𝑚󵄩1,𝑞 + |𝑢| + |V| Proof. Suppose that 𝑦=∑𝛽,𝑢 𝑐𝛼,V𝑥 𝑥 is an arbitrary element of 𝐾1,where𝑐𝛽,V ∈ F and 𝑐𝛽,V =0̸.Let𝑞∈N such + (𝛼 )+ (𝛽 −1) (𝛼) 𝑢 pad0 𝑚 pad0 𝑚 that 𝑞≥ht(𝑦).Let𝑥 𝑥 be a standard basis element of 𝐾. ℓ 󵄩 󵄩 ( 𝑦) 𝑞 (𝑥(𝛼)𝑥𝑢)= = ‖𝛼‖𝑞 +2‖𝛼‖1,𝑞 + |𝑢| + pad0 (𝛼𝑚)+󵄩𝛽−𝜀𝑚󵄩𝑞 By using Lemma 8 repeatedly, we see that ad 0. 󵄩 󵄩 +2󵄩𝛽−𝜀𝑚󵄩1,𝑞 + |V| + pad0 (𝛽𝑚 −1) Lemma 10. For 𝑖, 𝑗0 ∈𝐽 , the following statements hold. (𝛼) 𝑢 󵄩 󵄩 󵄩 󵄩 ≥ F𝑞 (𝑥 𝑥 )+󵄩𝛽󵄩 +2󵄩𝛽󵄩 −1+|V| + (𝛽𝑚)−1 󵄩 󵄩𝑞 󵄩 󵄩1,𝑞 pad0 (2𝜀𝑖) (1) 𝑥 ∈ nil(𝐾[0] ∩𝐾0). ≥ F (𝑥(𝛼)𝑥𝑢)+1. 𝑞 (2) If 𝜎(𝑖) = 𝜎(𝑗) and 𝑖 =𝑗̸,then𝑥𝑖𝑥𝑗 ∈ nil(𝐾[0] ∩𝐾0). (40) 󸀠 (3) If 𝜎(𝑖) ≠ 𝜎(𝑗) and 𝑖 =𝑗̸,then𝑥𝑖𝑥𝑗 ∈ nil(𝐾[0] ∩𝐾0).

(3) Similarly, we have (4) 𝑥𝑖𝑥𝑖󸀠 ∈ Nil(𝐾[0] ∩𝐾0).

(𝛽) V (𝛼) 𝑢 (1) 𝑥(𝛼)𝑥𝑢 𝐾 F𝑞 (𝐷𝑖 (𝑥 𝑥 )𝐷𝑖󸀠 (𝑥 𝑥 )) Proof. Let be a standard basis element of .A direct calculation shows that ((𝛽−𝜀 )+(𝛼−𝜀 )) V 𝑢 𝑖 𝑖󸀠 = F𝑞 (𝑥 𝑥 𝑥 ) (2𝜀 ) 𝑝 (𝛼) 𝑢 (2𝜀 ) 𝑝−1 (𝛼−𝜀 ) 𝑢 𝑖 𝑖 𝑖󸀠 (ad𝑥 ) (𝑥 𝑥 )=(ad𝑥 ) (𝜎 (𝑖) 𝑥𝑖𝑥 𝑥 ) 󵄩 󵄩 󵄩 󵄩 = 󵄩(𝛽 −𝑖 𝜀 )+(𝛼−𝜀𝑖󸀠 )󵄩𝑞 +2󵄩(𝛽 −𝑖 𝜀 )+(𝛼−𝜀𝑖󸀠 )󵄩1,𝑞 + |V| (43) 𝑝 𝑝 (𝛼−𝑝𝜀 ) 𝑢 𝑖󸀠 =𝜎(𝑖) 𝑥𝑖 𝑥 𝑥 =0. + |𝑢| + pad0 (𝛽𝑚 +𝛼𝑚) 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 (2) Since ad𝑥𝑖𝑥𝑗 = 𝜎(𝑖)(𝑥𝑗𝐷𝑖󸀠 +𝑥𝑖𝐷𝑗󸀠 ) and 𝑥𝑗𝐷𝑖󸀠 ∘ = 󵄩𝛽−𝜀𝑖󵄩𝑞 + 󵄩𝛼−𝜀𝑖󸀠 󵄩𝑞 +2󵄩𝛽−𝜀𝑖󵄩1,𝑞 +2󵄩𝛼−𝜀𝑖󸀠 󵄩1,𝑞 + |V| 𝑥𝑖𝐷𝑗󸀠 =𝑥𝑖𝐷𝑗󸀠 ∘𝑥𝑗𝐷𝑖󸀠 , it follows from the binomial theorem 𝑝 𝑝 𝑝 𝑝 𝑝 ( 𝑥 𝑥 ) = 𝜎(𝑖) (𝑥 𝐷 󸀠 +𝑥𝐷 󸀠 ) = 𝜎(𝑖) ((𝑥 𝐷 󸀠 ) + + |𝑢| + (𝛽𝑚)+ (𝛼𝑚) that ad 𝑖 𝑗 𝑗 𝑖 𝑖 𝑗 𝑗 𝑖 pad0 pad0 𝑝 (𝑥 𝐷 󸀠 ) )=0 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 𝑖 𝑗 . = 󵄩𝛽−𝜀󵄩 +2󵄩𝛽−𝜀󵄩 + 󵄩𝛼−𝜀󸀠 󵄩 +2󵄩𝛼−𝜀󸀠 󵄩 + |V| 󵄩 𝑖󵄩𝑞 󵄩 𝑖󵄩1,𝑞 󵄩 𝑖 󵄩𝑞 󵄩 𝑖 󵄩1,𝑞 (3) Since ad𝑥𝑖𝑥𝑗 = 𝜎(𝑖)(𝑥𝑗𝐷𝑖󸀠 −𝑥𝑖𝐷𝑗󸀠 ) and 𝑥𝑗𝐷𝑖󸀠 ∘𝑥𝑖𝐷𝑗󸀠 = 𝑝 𝑝 𝑝 𝑥𝑖𝐷𝑗󸀠 ∘𝑥𝑗𝐷𝑖󸀠 ,wehave(ad𝑥𝑖𝑥𝑗) = 𝜎(𝑖) (𝑥𝑗𝐷𝑖󸀠 −𝑥𝑖𝐷𝑗󸀠 ) = + |𝑢| + pad0 (𝛽𝑚)+pad0 (𝛼𝑚) 𝑝 𝑝 𝑝 𝜎(𝑖) ((𝑥𝑗𝐷𝑖󸀠 ) −(𝑥𝑖𝐷𝑗󸀠 ) )=0. 󵄩 󵄩 󵄩 󵄩 (2𝜀 ) (2𝜀 ) 󵄩 󵄩 󵄩 󵄩 𝑖 𝑖󸀠 ≥ 󵄩𝛽󵄩𝑞 +2󵄩𝛽󵄩1,𝑞 −1+‖𝛼‖𝑞 +2‖𝛼‖1,𝑞 −1+|V| + |𝑢| (4) It follows from (1) that 𝑥𝑖𝑥𝑖󸀠 = 𝜎(𝑖)⟦𝑥 ,𝑥 ⟧∈ Nil(𝐾[0] ∩𝐾0). + pad0 (𝛽𝑚)+pad0 (𝛼𝑚) Lemma 11. Suppose that 𝑛≥3. The following statements hold. (𝛽) V (𝛼) 𝑢 = F𝑞 (𝑥 𝑥 )+F𝑞 (𝑥 𝑥 )−2 (1) Let 𝑖, 𝑗, 𝑘 be distinct elements of 𝑌1,andlet𝑎, 𝑏 ∈ F be 2 2 such that 𝑎 +𝑏 =0.Then𝑦=𝑎𝑥𝑖𝑥𝑗 +𝑏𝑥𝑖𝑥𝑘 ∈ nil(𝐾). ≥ F (𝑥(𝛼)𝑥𝑢)+1. 𝑞 (2)𝑥𝑖𝑥𝑗 ∈ Nil(𝐾[0] ∩𝐾0) holds for all distinct 𝑖, 𝑗1 ∈𝑌 . (41) Proof. (1) Adirectcalculationshowsthatad𝑦= 𝑎𝑥𝑗𝐷𝑖 −𝑎𝑥𝑖𝐷𝑗 +𝑏𝑥𝑘𝐷𝑖 −𝑏𝑥𝑖𝐷𝑘. For simplicity, we denote (𝛽) V (𝛼) 𝑢 𝑎𝑥𝑗𝐷𝑖,−𝑎𝑥𝑖𝐷𝑗,𝑏𝑥𝑘𝐷𝑖,−𝑏𝑥𝑖𝐷𝑘 by 𝐴, 𝐵, 𝐶,𝐷,respectively. Lemma 8. Suppose that 𝑥 𝑥 ∈𝐾1, 𝑥 𝑥 ∈𝐾,and (𝛽) V (𝛼󸀠) 𝑢󸀠 Clearly, 𝑞≥max{1, ht(𝑥 𝑥 )}.Let𝑥 𝑥 be a nonzero summand (𝛽) V (𝛼) 𝑢 (𝛼󸀠) 𝑢󸀠 (𝛼) 𝑢 ⟦𝑥 𝑥 ,𝑥 𝑥 ⟧ F (𝑥 𝑥 )≥F (𝑥 𝑥 )+1 2 2 2 2 of .Then 𝑞 𝑞 . 𝐴 =𝐵 =𝐶 =𝐷 =0, 𝐴𝐶=𝐶𝐴=𝐵𝐷=𝐷𝐵=0.

(𝛼󸀠) 𝑢󸀠 (44) Proof. Adirectcalculationshowsthat𝑥 𝑥 fulfills the conditions of Lemma 7. Then

2 Given that 𝑞∈N,let (ad𝑦) = 𝐴𝐵 + 𝐵𝐴 + 𝐴𝐷 + 𝐷𝐴 + 𝐵𝐶 +𝐶𝐵𝐷𝐶𝐶𝐷. (45) ℓ𝑞 =𝑚(𝑞+1)(𝑝−1)+2𝑚𝑞(𝑝−1)+(𝑝−1)+𝑛+1. (42) By 𝐴𝐷𝐴 = 𝐷𝐴𝐷 = 𝐵𝐶𝐵 =𝐶𝐵𝐶0,weobtain

3 (𝛼) 𝑢 (𝛼) 𝑢 ( 𝑦) =𝐴𝐵𝐴+𝐴𝐵𝐶+𝐴𝐷𝐶+𝐵𝐴𝐵+𝐵𝐴𝐷+𝐵𝐶𝐷 Clearly, the inequality F𝑞(𝑥 𝑥 )<ℓ𝑞 holds for all 𝑥 𝑥 ∈ ad 𝐾 . +𝐶𝐵𝐴+𝐶𝐷𝐴+𝐶𝐷𝐶+𝐷𝐴𝐵+𝐷𝐶𝐵+𝐷𝐶𝐷.

Lemma 9. 𝐾1 ⊂ nil(𝐾). (46) 6 Journal of Applied Mathematics

Note that Conversely, Lemma 10 shows that spanF {𝑥𝑖𝑥𝑗 |𝑖,𝑗∈𝐽0}⊆ Nil(𝐾[0]),andLemma12 implies that span {𝑥𝑖𝑥𝑗 |𝑖∈𝐽0,𝑗 ∈ 𝐴𝐵𝐴2 =−𝑎 𝐴, 𝐵𝐴𝐵2 =−𝑎 𝐵, F 𝑌1}⊆Nil(𝐾[0]).Moreover,since𝑥𝑖𝑥𝑗 =⟦𝑥1𝑥𝑖,𝑥1󸀠 𝑥𝑗⟧ for 𝑖, 𝑗 ∈𝑌 {𝑥 𝑥 |𝑖,𝑗∈𝑌}⊆ (𝐾 ) 𝐶𝐷𝐶 =−𝑏2𝐶, 𝐷𝐶𝐷2 =−𝑏 𝐷, all 1,wehavespanF 𝑖 𝑗 1 Nil [0] . Therefore spanF {𝑥𝑖𝑥𝑗 |𝑖,𝑗∈𝐽}⊆Nil(𝐾[0]). (47) (2) (𝐾 )⊆ (𝐾 ) 𝑎2𝐴=𝐶𝐷𝐴+𝐴𝐷𝐶,2 𝑎𝐵=𝐵𝐶𝐷+𝐷𝐶𝐵, It is clear that Nil [0] Nil 0 ,whichcombined with Lemma 9 yields Nil(𝐾[0])+𝐾1 ⊆ Nil(𝐾0). 2 2 𝑦=𝑦 +𝑦 𝑏 𝐶 = 𝐴𝐵𝐶 + 𝐶𝐵𝐴,𝑏 𝐷 = 𝐵𝐴𝐷 + 𝐷𝐴𝐵. On the other hand, suppose that [0] 1 is an arbitrary element of nil(𝐾0),where𝑦[0] ∈𝐾[0] and 𝑦1 ∈𝐾1. 𝑦 ∈ (𝐾 ) 𝑦=𝑦 + It follows that By Lemma 5,wehave [0] Nil [0] , and hence [0] 𝑦1 ∈ Nil(𝐾[0])+𝐾1.SinceNil(𝐾[0])+𝐾1 isasubalgebraof𝐾, 𝐴𝐵𝐴+𝐶𝐷𝐴+𝐴𝐷𝐶=0, 𝐵𝐴𝐵+𝐵𝐶𝐷+𝐷𝐶𝐵=0, it follows that Nil(𝐾0)⊆Nil(𝐾[0])+𝐾1. (3) (1) (𝐾 ∩𝐾)⊆ (𝐾 )∩𝐾 = 𝐶𝐷𝐶+𝐴𝐵𝐶+𝐶𝐵𝐴=0, 𝐷𝐶𝐷+𝐵𝐴𝐶+𝐷𝐴𝐵=0, By ,weseethatNil [0] 0 Nil [0] 0 spanF {𝑥𝑖𝑥𝑗 |𝑖,𝑗∈𝐽, 𝜇(𝑖) = 𝜇(𝑗)}. The reverse inclusion (48) follows from Lemmas 10 and 11. 3 (4) Clearly the statement holds when 𝑛=1.Nowwe thus proving that (ad𝑦) =0. 𝑛=2 (1) 𝑦= 2 consider the case .By ,wecansupposethat (2) Let 𝑎∈F such that 𝑎 =−1.Sincechar(F)>2,we 𝑎𝑥 𝑥 + ∑ 𝑎 𝑥 𝑥 (𝐾 ∩ 2 𝑚+1 𝑠 𝑖,𝑗∈𝐽0 𝑖𝑗 𝑖 𝑗 is an arbitrary element of nil [0] have 𝑎 −1=−2=0̸.Let𝑘∈𝑌1 \ {𝑖, 𝑗}.Then(1) yields 𝐾0),where𝑎,𝑖𝑗 𝑎 ∈ F.If𝑎 =0̸, a direct calculation shows that ( 𝑦)2𝑡(𝑥 )=(−1)𝑡𝑎2𝑡𝑥 =0̸ 𝑡∈N 𝑦1 =𝑎𝑥𝑖𝑥𝑗 +𝑥𝑖𝑥𝑘 ∈ Nil (𝐻[0] ∩𝐻), ad 𝑠 𝑠 for all , contradicting that 0 𝑦 𝑎=0 𝑦∈ {𝑥 𝑥 |𝑖,𝑗∈𝐽} (49) is ad-nilpotent. Hence and spanF 𝑖 𝑗 0 , 𝑦2 =𝑥𝑖𝑥𝑗 +𝑎𝑥𝑖𝑥𝑘 ∈ Nil (𝐻[0] ∩𝐻0). proving nil(𝐾[0] ∩𝐾0)⊆spanF {𝑥𝑖𝑥𝑗 |𝑖,𝑗∈𝐽0}.Since spanF {𝑥𝑖𝑥𝑗 |𝑖,𝑗∈𝐽0} is a subalgebra of 𝐾,itfollowsthat Hence 𝑥𝑖𝑥𝑗 = −(1/2)(𝑎𝑦1 −𝑦2)∈Nil(𝐻[0] ∩𝐻0). Nil(𝐾[0] ∩𝐾0)⊆spanF {𝑥𝑖𝑥𝑗 |𝑖,𝑗∈𝐽0}. The reverse inclusion follows from Lemma 10. Lemma 12. 𝑖∈𝐽 𝑗∈𝑌 𝑥 𝑥 ∈ (𝐾 ∩𝐾 ) Let 0 and 1.Then 𝑖 𝑗 nil [0] 1 . (5) is completely analogous to the proof of (2).

Proof. Adirectcalculationshowsthat Let 𝜌 be the corresponding representation with respect to 𝐾[0]-module 𝐾[−1];thatis,𝜌(𝑦) = ad𝑦|𝐾 , ∀𝑦 ∈[0] 𝐾 .Itis ad𝑥𝑖𝑥𝑗 =𝜎(𝑖) 𝑥𝑗𝐷𝑖󸀠 −𝑥𝑖𝐷𝑗, [−1] easy to see that 𝜌 is faithful. For 𝑦∈𝐾[0], we also denote by (50) 2 𝜌(𝑦) the matrix of 𝜌(𝑦) relative to the fixed ordered F-basis (ad𝑥𝑖𝑥𝑗) =−𝜎(𝑖) (𝑥𝑖𝐷𝑗 ∘𝑥𝑗𝐷𝑖󸀠 +𝑥𝑗𝐷𝑖󸀠 ∘𝑥𝑖𝐷𝑗). as follows: Since {𝑥1,𝑥2,...,𝑥𝑚−1,𝑥𝑚+1,...,𝑥𝑠} . (53) (𝑥𝑖𝐷𝑗 ∘𝑥𝑗𝐷𝑖󸀠 )∘(𝑥𝑗𝐷𝑖󸀠 ∘𝑥𝑖𝐷𝑗) (2𝑟, 𝑛) (2𝑟+ (51) Denote by gl the general linear Lie superalgebra of 𝑛)×(2𝑟+𝑛) F 𝑒 (𝑠 − 1) × (𝑠 − =0=(𝑥𝑗𝐷𝑖󸀠 ∘𝑥𝑖𝐷𝑗)∘(𝑥𝑖𝐷𝑗 ∘𝑥𝑗𝐷𝑖󸀠 ), matrices over .Let 𝑖𝑗 denote the 1) matrix whose (𝑖, 𝑗)-entry is 1 and 0 elsewhere, and 𝐺= 0𝐼 ( 𝑟 ) 𝐼 𝑟×𝑟 (2𝑟, F) we have −𝐼𝑟 0 ,where 𝑟 is unit matrix. Let sp be the Lie 2𝑝 algebra consisting of all 2𝑟 × 2𝑟 matrices 𝐴 over F satisfying ( 𝑥 𝑥 ) 𝑇 𝑇 ad 𝑖 𝑗 𝐴 𝐺+𝐺𝐴=,where 0 𝐴 is the transpose of 𝐴.SetK = 𝑝 𝑝 (52) L ⊕ F𝐼𝑠−1;here =−𝜎(𝑖) ((𝑥𝑖𝐷𝑗 ∘𝑥𝑗𝐷𝑖󸀠 ) +(𝑥𝑗𝐷𝑖󸀠 ∘𝑥𝑖𝐷𝑗) )=0. 𝐴𝐵 L ={( )∈ (2𝑟, 𝑛) |𝐴∈ (2𝑟, F) ,𝐵𝑇𝐺+𝐶=0, 𝐶𝐷 gl sp Lemma 13. (1) Nil(𝐾[0])=spanF {𝑥𝑖𝑥𝑗 |𝑖,𝑗∈𝐽}. (2) (𝐾 )= (𝐾 )+𝐾 Nil 0 Nil [0] 1. 𝐷 anti-symmetric}. (3) If 𝑛≥3,thenNil(𝐾[0] ∩𝐾0)=spanF {𝑥𝑖𝑥𝑗 |𝑖,𝑗∈ 𝐽, 𝜇(𝑖) = 𝜇(𝑗)}. (54) (4) If 𝑛≤2,thenNil(𝐾[0] ∩𝐾0)=spanF {𝑥𝑖𝑥𝑗 |𝑖,𝑗∈𝐽0}. Lemma 14. (1) 𝜌(𝐾 )=K (5) Nil(𝐾0 ∩𝐾0)=Nil(𝐾[0] ∩𝐾0)+𝐾1 ∩𝐾0. [0] . (2) If 𝑦∈nil(𝐾[0]),then𝜌(𝑦) is a nilpotent matrix. Proof. (1) Let 𝑦=𝑎𝑚𝑥𝑚 +∑𝑖,𝑗∈𝐽 𝑎𝑖𝑗 𝑥𝑖𝑥𝑗 be an arbitrary 𝑡 (1) 𝑖, 𝑗 ∈𝐽 element of nil(𝐾[0]).Supposethat𝑎𝑚 =0̸.Since(ad𝑦) (1) = Proof. For , a direct calculation shows that 𝑡 (−2𝑎𝑚) =0̸and ∀𝑡 ∈ N,itfollowsthat𝑦 is not ad-nilpotent, 𝜇(𝑖)(𝜇(𝑖)+𝜇(𝑗)) ⟦𝑥𝑖𝑥𝑗,𝑥𝑘⟧=𝜎(𝑖)(−1) 𝛿𝑖󸀠𝑘𝑥𝑗 a contradiction. Hence 𝑎𝑚 =0, and therefore nil(𝐾[0])⊆ (55) spanF {𝑥𝑖𝑥𝑗 |𝑖,𝑗∈𝐽}.NotingthatspanF {𝑥𝑖𝑥𝑗 |𝑖,𝑗∈𝐽}is a 𝜇(𝑗)(𝜇(𝑖)+𝜇(𝑗))+𝜇(𝑖)𝜇(𝑗) + 𝜎 (𝑗) (−1) 𝛿𝑗󸀠𝑘𝑥𝑖. subalgebra of 𝐾,weobtainNil(𝐾[0])⊆spanF {𝑥𝑖𝑥𝑗 |𝑖,𝑗∈𝐽}. Journal of Applied Mathematics 7

Therefore Assume that there exists some 𝑎𝑖𝑗 =0̸.Fordistinct𝑘, 𝑙 ∈ 𝑁\{𝑖,𝑗} 𝑎 =0 𝜌(𝑥𝑥 )=𝜎(𝑖)(−1)𝜇(𝑖)+𝜇(𝑖)𝜇(𝑗)𝑒 + 𝜎 (𝑗) (−1)𝜇(𝑗)𝑒 , ,wehave 𝑘𝑙 by (60).Thenwecanwritethe 𝑖 𝑗 𝑗̂𝑖̂󸀠 ̂𝑖𝑗̂󸀠 (56) following: where 𝐴=∑𝑎 𝐸 + ∑𝑎 𝐸 − ∑𝑎 𝐸 − ∑𝑎 𝐸 . 𝑖𝑡 𝑖𝑡 𝑗𝑡 𝑗𝑡 𝑖𝑡 𝑡𝑖 𝑗𝑡 𝑡𝑗 (64) ̂ 𝑙 if 𝑙∈𝐽0, 𝑡∈𝑁 𝑡∈𝑁 𝑡∈𝑁 𝑡∈𝑁 𝑙={ (57) 𝑙−1 if 𝑙∈𝑌1. Adirectcalculationshowsthat

Since ⟦𝑥𝑚,𝑥𝑘⟧=−𝑥𝑘, ∀𝑘 ∈ 𝐽,weseethat 2 2 2 2 0=Γ([𝐴,𝑘𝑙 𝐸 −𝐸𝑙𝑘]) =𝑎𝑖𝑙 +𝑎𝑗𝑙 +𝑎𝑖𝑘 +𝑎𝑗𝑘. (65) 𝜌 (𝑥𝑚) =−𝐼𝑠−1. (58) 𝑎2 +𝑎2 =𝑎2 +𝑎2 𝑎2 +𝑎2 =0 From (56)and(58), we can easily verify (1). Since 𝑖𝑙 𝑗𝑘 𝑗𝑙 𝑖𝑘 by (63), we obtain 𝑖𝑙 𝑗𝑘 . 𝑎2 =𝑎2 +𝑎2 =𝑎2 +𝑎2 =0 (2) follows from the definition of 𝜌. Hence 𝑖𝑗 𝑖𝑗 𝑘𝑙 𝑖𝑙 𝑗𝑘 by (63), contradicting the assumption that 𝑎𝑖𝑗 =0̸. If 𝐴=(𝑎𝑖𝑗 ) is an 𝑛×𝑛antisymmetric matrix over F,we 2 2 2 write Γ(𝐴) to denote ∑1≤𝑖<𝑗≤𝑛 𝑎𝑖𝑗 .Astr(𝐴 )=−2∑1≤𝑖<𝑗≤𝑛 𝑎𝑖𝑗 , Case 2 (𝑛 = 4).Notethat(60)and(63)hold;thatis, it is clear that if 𝐴 is a nilpotent matrix then Γ(𝐴) = 0. 𝑎12𝑎34 =𝑎13𝑎24 =𝑎14𝑎23 =0, Lemma 15. Suppose that 𝑛≥3.Let𝐴=(𝑎𝑖𝑗 ) be an (66) 𝑎2 +𝑎2 =𝑎2 +𝑎2 =𝑎2 +𝑎2 . antisymmetric matrix over F of order 𝑛.If𝐴 satisfies the 12 34 13 24 23 14 following properties: 2 2 Moreover, since 0 = Γ(𝐴) − Γ([𝐴,12 𝐸 −𝐸21]) = 𝑎 +𝑎 ,we Γ(𝐴) = 0 12 34 (1) ; see that 𝐴=0. (2) Γ([𝐴, 𝐵]) =0 holds for every 𝑛×𝑛antisymmetric (𝑛 = 3) 0 = Γ(𝐴) = 𝑎2 +𝑎2 +𝑎2 Γ([𝐴, 𝐸 − matrix 𝐵. Case 3 .Since 12 23 13 and 12 𝐸 ]) = 𝑎2 +𝑎2 𝑎 =0 𝑎 =𝑎 = 𝐴=0 21 23 13,itfollowsthat 12 .Similarly 23 13 Then, . 0. Proof. Let 𝑁={1,2,...,𝑛}.Threecasesariseasfollows. Lemma 16. Let 𝑦 be a nonzero element of nil(𝐾[0] ∩𝐾0).Then there exists 𝑧∈𝐾[0] ∩𝐾 such that ⟦𝑦, 𝑧⟧ is not 𝑎𝑑-nilpotent. Case 1 (𝑛 > 4).Bytheproperty(2) of 𝐴,weseethat 0 Proof. By Lemma 13,wecansupposethat Γ([𝐴,𝐸𝑖𝑗 −𝐸𝑗𝑖 +𝐸𝑘𝑙 −𝐸𝑙𝑘]) − Γ ([𝐴,𝑖𝑗 𝐸 −𝐸𝑗𝑖]) (59) (2𝜀 ) − Γ ([𝐴,𝑘𝑙 𝐸 −𝐸𝑙𝑘]) = 0 𝑦=∑𝑎 𝑥 𝑙 + ∑ 𝑏 𝑥 𝑥 + ∑ 𝑐 𝑥 𝑥 , 𝑙 𝑙𝑡 𝑙 𝑡 𝑙𝑡 𝑙 𝑡 (67) 𝑙∈𝐽 𝑙,𝑡∈𝐽 ,𝑙<𝑡 𝑙,𝑡∈𝑌 ,𝑙<𝑡 holds for all 𝑖, 𝑗, 𝑘, 𝑙∈𝑁. Therefore, if 𝑖<𝑗<𝑘<𝑙,adirect 0 0 1 calculationshowsthattheleft-handsideof(59)isequalto where 𝑎𝑙,𝑏𝑙𝑡,𝑐𝑙𝑡 ∈ F.Threecasesariseasfollows. 4𝑎𝑖𝑙𝑎𝑗𝑘 −4𝑎𝑖𝑘𝑎𝑗𝑙.Hence4𝑎𝑖𝑙𝑎𝑗𝑘 −4𝑎𝑖𝑘𝑎𝑗𝑙 =0.Similarly,if𝑖< (2𝜀 ) 𝑘<𝑗<𝑙,then−4𝑎𝑖𝑙𝑎𝑘𝑗 −4𝑎𝑖𝑘𝑎𝑗𝑙 =0,andif𝑖<𝑘<𝑙<𝑗, 𝑖󸀠 Case 1. There exists some 𝑎𝑖 =0̸.Let𝑧=𝑥 .Then𝑦, 𝑧 = then −4𝑎𝑖𝑙𝑎𝑘𝑗 +4𝑎𝑖𝑘𝑎𝑙𝑗 =0.Thus,forall𝑖1 <𝑖2 <𝑖3 <𝑖4,we 𝜎(𝑖)𝑎𝑖𝑥𝑖𝑥𝑖󸀠 +ℎ,where𝑥𝑖 does not appear in the expression see that 𝑎𝑖 𝑖 𝑎𝑖 𝑖 =𝑎𝑖 𝑖 𝑎𝑖 𝑖 =−𝑎𝑖 𝑖 𝑎𝑖 𝑖 =−𝑎𝑖 𝑖 𝑎𝑖 𝑖 .Then ℓ ℓ 1 3 2 4 1 4 2 3 1 2 3 4 1 3 2 4 of ℎ.Notingthat(ad⟦𝑦, 𝑧⟧) (𝑥𝑖󸀠 )=𝑎𝑖 𝑥𝑖󸀠 for all ℓ∈N,we 𝑎 𝑎 =𝑎 𝑎 =𝑎 𝑎 =0. conclude that ⟦𝑦, 𝑧⟧ is not ad-nilpotent. 𝑖1𝑖2 𝑖3𝑖4 𝑖1𝑖3 𝑖2𝑖4 𝑖1𝑖4 𝑖2𝑖3 (60)

𝑛 2 Case 2.All𝑎𝑙 =0, and there exists 𝑏𝑖𝑗 =0̸.Let𝑧=𝑥𝑖󸀠𝑥𝑗󸀠 . Denote 𝜂𝑖 =∑𝑡=1 𝑎𝑖𝑡 for 𝑖∈𝑁.If𝑖 =𝑗̸, a direct calculation ⟦𝑦, 𝑧⟧ = 𝜎(𝑗)𝑎 𝑥 𝑥 󸀠 + shows that Adirectcalculationshowsthat 𝑖𝑗 𝑖 𝑖 ℎ,where𝑥𝑖 does not appear in the expression of ℎ.As 0=Γ([𝐴,𝐸 −𝐸 ]) = 𝜂 +𝜂 −2𝑎2 . ℓ ℓ ℓ ℓ 𝑖𝑗 𝑗𝑖 𝑖 𝑗 𝑖𝑗 (61) (ad⟦𝑦, 𝑧⟧) (𝑥𝑖󸀠 )=𝜎(𝑖)𝜎(𝑗) 𝑎𝑖𝑗 𝑥𝑖󸀠 and ∀ℓ ∈ N,weseethat ⟦𝑦, 𝑧⟧ is not ad-nilpotent. Let 𝑘∈𝑁\{𝑖,𝑗}.Then

2 2 Case 3.All𝑎𝑙 =0and 𝑏𝑙𝑡 =0.So𝑦∈spanF {𝑥𝑖𝑥𝑗 |𝑖,𝑗∈𝑌1}. 𝜂𝑖 +𝜂𝑘 −2𝑎𝑖𝑘 =0, 𝜂𝑗 +𝜂𝑘 −2𝑎𝑗𝑘 =0. (62) This can happen only if 𝑛≥3by Lemma 13(4).Notethat 2 2 2 𝜌(𝑦) is an antisymmetric nilpotent matrix. Then Lemma 15 From the equalities above, we see that 𝜂𝑖 =𝑎𝑖𝑗 +𝑎𝑖𝑘 −𝑎𝑗𝑘 provides an element 𝑧 of span {𝑥𝑖𝑥𝑗 |𝑖,𝑗∈𝑌1} such that holds for all distinct 𝑖, 𝑗, 𝑘∈𝑁.Pick𝑙∈𝑁\{𝑖,𝑗,𝑘}.Then F 2 2 2 2 2 2 2 ⟦𝜌(𝑦), 𝜌(𝑧)⟧ is not a nilpotent matrix. Hence ⟦𝑦, 𝑧⟧ is not ad- 𝜂𝑖 =𝑎 +𝑎 −𝑎 .Hence𝑎 +𝑎 =𝑎 +𝑎 holds for all 𝑖𝑗 𝑖𝑙 𝑗𝑙 𝑖𝑙 𝑗𝑘 𝑖𝑘 𝑗𝑙 nilpotent. distinct 𝑖, 𝑗, 𝑘, 𝑙∈𝑁.Itfollowsthat 𝑎2 +𝑎2 =𝑎2 +𝑎2 =𝑎2 +𝑎2 𝑖1𝑖2 𝑖3𝑖4 𝑖1𝑖3 𝑖2𝑖4 𝑖2𝑖3 𝑖1𝑖4 (63) 4. Proofs of Theorems holds for all 𝑖1 <𝑖2 <𝑖3 <𝑖4. Proof of Theorem 1. We proceed in several steps. 8 Journal of Applied Mathematics

(I) Nil(𝐾0)=Nil(𝐾0 ∩𝐾0).Supposethat𝑦 is an arbitrary The reverse inclusion follows from the fact that 𝐾1 ⊂𝐾−1. element of nil(𝐾0).Itfollowsfrom(2) and (3) of Lemma 5 (V) ⟦𝐾1,𝐾1 ∩𝐾0⟧=𝐾0 ∩𝐾1.Itsufficestoshowthat 𝑦∈𝐾 𝑦∈ (𝐾 ) 𝑦∈ (𝛼) 𝑢 that 0,whichcombinedwith nil 0 yields 𝐾0 ∩𝐾1 ⊆⟦𝐾1,𝐾1 ∩𝐾0⟧.Supposethat𝑥 𝑥 is an arbitrary nil(𝐾0 ∩𝐾),thusprovingnil(𝐾 )⊆nil(𝐾0 ∩𝐾).Hence 𝑢 V (𝛼+𝜀𝑚) V 0 0 0 basis element of 𝐾0 ∩𝐾 with 𝑥 =𝑥𝑖𝑥 .Notethat𝑥 𝑥 ∈ (𝐾 )⊆ (𝐾 ∩𝐾) 1 Nil 0 Nil 0 0 . (𝛼) 𝑢 (𝛼+𝜀𝑚) V (𝛼) 𝑢 𝐾1∩𝐾 .Since𝑥 𝑥 =⟦𝑥𝑖,𝑥 𝑥 ⟧,itfollowsthat𝑥 𝑥 ∈ The reverse inclusion is clear. 0 ⟦𝐾1,𝐾1 ∩𝐾0⟧, as desired. (II) 𝐾0 ∩𝐾0 = Nor𝐾 (Nil(𝐾0)).Wefirstprove𝐾0 ∩𝐾0 ⊆ 0 It follows from (II) and (V) that 𝐾0 =𝐾0 ∩𝐾0 +𝐾0 ∩𝐾1 Nor𝐾 (Nil(𝐾0)). It follows from (I) that 0 is invariant under automorphisms of 𝐾. By (III) and (IV), we obtain that 𝐾1 =𝐾1 ∩𝐾0 +𝐾1 ∩𝐾1 is invariant. Therefore Nor𝐾 (Nil (𝐾0)) = Nor𝐾 (Nil (𝐾0 ∩𝐾0)) . (68) 0 0 𝐾−1 ={𝑥∈𝐾|⟦𝑥,𝐾1⟧⊆𝐾0} is invariant. By the transitivity Therefore of 𝐾,weconcludethat

(𝐾 ∩𝐾) ⊆ ( (𝐾 ∩𝐾)) = ( (𝐾 )) . 𝐾𝑖+1 = {𝑥∈𝐾𝑖 | ⟦𝑥,−1 𝐾 ⟧ ⊆𝐾𝑖} ,∀𝑖≥0. Nil 0 0 Nor𝐾0 Nil 0 0 Nor𝐾0 Nil 0 (71) (69) Hence the natural filtration of 𝐾 is invariant under automor- 𝐾 By formula (12), we see that ⟦𝑥𝑚,𝐾[0]⟧=0and ⟦𝑥𝑚,𝐾1 ∩ phisms of . 𝐾 ⟧⊆𝐾1 ∩𝐾,proving 󸀠 0 0 Proof of Theorem 2. Let 𝜑 : 𝐾(2𝑟 + 1, 𝑛) →𝐾(2𝑟 + 󸀠 󸀠 𝑥 ∈ ( (𝐾 )) . 1, 𝑛 ) be an isomorphism of Lie superalgebras. Let 𝐾 and 𝐾 𝑚 Nor𝐾0 Nil 0 (70) 󸀠 󸀠 denote 𝐾(2𝑟 + 1, 𝑛) and 𝐾(2𝑟 +1,𝑛),respectively.Since 𝑛≥3 (3) (5) 𝜑(𝐾 )=𝐾󸀠 𝜑( (𝐾 )) = (𝐾󸀠 ) In the case of ,by and of Lemma 13,weobtain 0 0 and Nil 0 Nil 0 ,itfollowsthat 󸀠 𝐾0 ∩𝐾 = (𝐾0 ∩𝐾)+F𝑥𝑚 0 Nil 0 .By(69)and(70), we have 𝜑(Nor𝐾 (Nil(𝐾0))) = Nor𝐾󸀠 (Nil(𝐾 )). By (II) in the proof of 0 0 0 𝐾0 ∩𝐾0 ⊆ Nor𝐾 (Nil(𝐾0)).Inthecaseof𝑛=2,by(4) and (5) 0 Theorem 1,wehave of Lemma 13,wehave𝐾0∩𝐾0 = Nil(𝐾0∩𝐾0)+F𝑥𝑚+F𝑥𝑚+1𝑥𝑠. Note that 𝑥𝑚+1𝑥𝑠 ∈ Nor𝐾 (Nil(𝐾0)),whichcombinedwith 𝜑(𝐾 ∩𝐾)=𝜑( ( (𝐾 ))) 0 0 0 Nor𝐾0 Nil 0 (69)and(70)yields𝐾0 ∩𝐾0 ⊆ Nor𝐾 (Nil(𝐾0)).Inthecase 0 󸀠 󸀠 󸀠 (72) = 󸀠 ( (𝐾 )) = 𝐾 ∩𝐾. of 𝑛=1,by(4) and (5) of Lemma 13,wehave𝐾0 ∩𝐾0 = Nor𝐾 Nil 0 0 0 (𝐾 ∩𝐾 )+F𝑥 𝐾 ∩𝐾 ⊆ ( (𝐾 )) 0 Nil 0 0 𝑚.Hence 0 0 Nor𝐾0 Nil 0 by (69) and (70). Consequently Conversely, suppose that 𝑦=𝑦[−2] +𝑦−1 ∈ 𝜑({𝑦∈nil (𝐾0)|⟦𝑦,𝐾0 ∩𝐾0⟧⊆nil (𝐾0)}) Nor𝐾 (Nil(𝐾0)),where𝑦[−2] ∈𝐾[−2], 𝑦−1 ∈𝐾−1.If𝑦[−2] =0̸, 0 (73) ⟦𝑦, 𝑥 𝑥 ⟧=2𝑦 𝑥 +ℎ ∉ (𝐾 ) (5) 󸀠 󸀠 󸀠 󸀠 then 1 𝑚 [−2] 1 Nil 0 by Lemma 13 , ={𝑦∈nil (𝐾 )|⟦𝑦,𝐾 ∩𝐾⟧⊆nil (𝐾 )} . 𝑦∈ ( (𝐾 )) ℎ∈𝐾 0 0 0 0 contradicting Nor𝐾0 Nil 0 ,where 0.Hence 𝑦[−2] =0and we can write 𝑦=𝑦[−1] +𝑦0,where𝑦[−1] = Applying (III) in the proof of Theorem 1,weseethat𝜑(𝐾1 ∩ ∑ 𝑎 𝑥 ∈𝐾 𝑎 ∈ F 𝑦 ∈𝐾 𝑎 =0̸ 𝐾 )=𝐾󸀠 ∩𝐾󸀠 𝑖∈𝐽0 𝑖 𝑖 [−1], 𝑖 , 0 0.Ifthereissome 𝑗 ,then 0 1 0, which combined with (V) yields (2𝜀 󸀠 ) ⟦𝑦, 𝑥 𝑗 ⟧=𝜎(𝑗)𝑎𝑥 󸀠 +ℎ ℎ∈𝐾 (5) 𝑗 𝑗 ,where 0.Lemma13 shows 𝜑(𝐾 ∩𝐾)=𝜑(⟦𝐾,𝐾 ∩𝐾⟧) = ⟦𝐾󸀠 ,𝐾󸀠 ∩𝐾󸀠 ⟧ (2𝜀 ) 0 1 1 1 0 1 1 0 ⟦𝑦, 𝑥 𝑗󸀠 ⟧∉ (𝐾 ) 𝑦∈ ( (𝐾 )) that Nil 0 , contradicting Nor𝐾0 Nil 0 . (74) 𝑦 =0 𝑦∈𝐾 ∩𝐾 ( (𝐾 )) ⊆ =𝐾󸀠 ∩𝐾󸀠 . Hence [−1] and 0 0,provingNor𝐾0 Nil 0 0 1 𝐾0 ∩𝐾0. 󸀠 (III) Let 𝑀={𝑦∈nil(𝐾0)|⟦𝑦,𝐾0 ∩𝐾0⟧⊆nil(𝐾0)}. It follows from (72)and(74)that𝜑(𝐾0)=𝐾0. Therefore 𝜑 Then 𝑀=𝐾1 ∩𝐾0.Supposethat𝑦=𝑦[0] +𝑦1 is an arbitrary induces an isomorphism of Z2-graded spaces 𝜑:𝐾/𝐾̃ 0 → 󸀠 󸀠 󸀠 element of 𝑀,where𝑦[0] ∈𝐾[0] ∩𝐾0 and 𝑦1 ∈𝐾1 ∩𝐾0.If 𝐾 /𝐾0. A comparison of dimensions shows that 𝑟=𝑟 and 󸀠 𝑦[0] =0̸,since𝑦[0] ∈ nil(𝐾[0]) by Lemma 5(4),thenLemma16 𝑛=𝑛.Theconverseimplicationisclear. provides an element 𝑧∈𝐾[0] ∩𝐾 such that ⟦𝑦[0],𝑧⟧is not ad- 0 𝜙| =𝜓| nilpotent. Hence ⟦𝑦, 𝑧⟧ is not ad-nilpotent by Lemma 5(4), Proof of Theorem 3. It suffices to prove that 𝐾[−1] 𝐾[−1] 𝜙=𝜓 𝜑(1) = 𝜑(⟦𝑥 ,𝑥 󸀠 ⟧) = contradicting 𝑦∈𝑀. Therefore 𝑦[0] =0and 𝑦∈𝐾1 ∩𝐾0, implies that .Since 1 1 𝑀⊆𝐾 ∩𝐾 𝜓(⟦𝑥 ,𝑥 󸀠 ⟧) = 𝜓(1) 𝜑| =𝜓| thus proving 1 0. 1 1 ,itfollowsthat 𝐾[−2] 𝐾[−2] .Weuse On the other hand, since ⟦𝐾1 ∩𝐾0,𝐾0 ∩𝐾0⟧⊆𝐾1 ∩𝐾0 ⊆ induction on ℓ to show that nil(𝐾0 ∩𝐾0) by Lemma 9,itfollowsthat𝐾1 ∩𝐾0 ⊆𝑀. 𝜑|𝐾 =𝜓|𝐾 ,∀ℓ≥−1. (75) (IV) Let 𝑄={𝑦∈𝐾1 |⟦𝑦,𝐾1⟧⊆𝐾0 ∩𝐾0}.Then𝑄= [ℓ] [ℓ] 𝐾 ∩𝐾 𝑦=𝑦 +𝑦 ∈𝑄 𝑦 = 1 1.Supposethat [−1] 0 ,where [−1] Assume that ℓ≥0and (75)holdsforℓ−1.Supposethat ∑𝑖∈𝑌 𝑎𝑖𝑥𝑖 ∈𝐾[−1] ∩𝐾1 and 𝑦0 ∈𝐾0 ∩𝐾1.Ifthereissome 1 𝑦∈𝐾[ℓ] and let 𝑧 = 𝜑(𝑦) .Wewanttoprove− 𝜓(𝑦) 𝑧= 𝑎𝑗 =0̸,then⟦𝑦,𝑗 𝑥 ⟧=−𝑎𝑗 +⟦𝑦0,𝑥𝑗⟧∉𝐾−1 ∩𝐾0, and hence 0. The induction hypothesis yields that 𝜑(𝑥𝑖) = 𝜓(𝑥𝑖) and ⟦𝑦,𝑗 𝑥 ⟧∉𝐾0 ∩𝐾0, contradicting 𝑦∈𝑄. Therefore 𝑦[−1] =0, 𝜑(⟦𝑦,𝑖 𝑥 ⟧) = 𝜓(⟦𝑦,𝑖 𝑥 ⟧), 𝑖∈𝐽. Therefore, andwecanwrite𝑦=𝑦[0]+𝑦1,where𝑦[0] =∑𝑖∈𝐽 ,𝑗∈𝑌 𝑎𝑖𝑗 𝑥𝑖𝑥𝑗 ∈ 0 1 ⟦𝑧, 𝜑 (𝑥 )⟧ = ⟦𝜑 (𝑦) − 𝜓 (𝑦) ,𝜑(𝑥 )⟧ 𝐾[0] ∩𝐾1 and 𝑦1 ∈𝐾1 ∩𝐾1. If there exists some 𝑎𝑙𝑡 =0̸,then 𝑖 𝑖 ⟦𝑦, 𝑥 ⟧=−∑ 𝑎 𝑥 +⟦𝑦,𝑥⟧∉𝐾 ∩𝐾 (76) 𝑡 𝑖∈𝐽0 𝑖𝑡 𝑖 1 𝑡 0 0, a contradiction =𝜑(⟦𝑦,𝑥𝑖⟧) − 𝜓 (⟦𝑦,𝑖 𝑥 ⟧) = 0. which yields 𝑦[0] =0and 𝑦∈𝐾1 ∩𝐾1. Journal of Applied Mathematics 9

Since 𝜑(𝐾0)=𝐾0 and 𝜑(𝐾−1)=𝐾−1 by Theorem 1, [14] Y. Zhang, “Finite-dimensional Lie superalgebras of Cartan type 𝜑 induces an automorphism 𝜑̃ of the Z2-graded space over fields of prime characteristic,” Chinese Science Bulletin,vol. 𝐾−1/𝐾0. Consequently there exists a homogeneous basis 42,no.9,pp.720–724,1997. {ℎ1,...,ℎ𝑚−1,ℎ𝑚+1,...,ℎ𝑠} of 𝐾[−1] such that (𝑥𝑖)≡ℎ𝑖(mod [15] Q. Mu and Y.Zhang, “Infinite-dimensional modular special odd 𝐾0). Thus there exist 𝑔𝑖 ∈𝐾0 such that 𝜑(𝑥𝑖)=ℎ𝑖 −𝑔𝑖, 𝑖∈𝐽. contact superalgebras,” Journal of Pure and Applied Algebra,vol. Therefore, (76)showsthat⟦𝑧,𝑖 ℎ ⟧=⟦𝑧,𝑔𝑖⟧ for all 𝑖∈𝐽.As 214, no. 8, pp. 1456–1468, 2010. 𝑧 = 𝜑(𝑦)−𝜓(𝑦)0 ∈𝐾 by Theorem 1,itcanbedecomposedinto 𝑡 𝑧=∑𝑗=0 𝑧[𝑗],where𝑧[𝑗] ∈𝐾[𝑗].Notingthat⟦𝑧[0],ℎ𝑖⟧∈𝐾[−1] and ⟦𝑧,𝑖 𝑔 ⟧∈𝐾0,weobtain⟦𝑧[0],ℎ𝑖⟧=0for all 𝑖∈𝐽,and hence 𝑧[0] =0since 𝐾 is transitively graded. By induction, we conclude that 𝑧[𝑗] =0, 𝑗 = 0,1,...,𝑡.Hence,𝑧=0,as desired.

Conflict of Interests The author declares that there is no conflict of interests regarding the publication of this paper.

References

[1] R. E. Block and R. L. Wilson, “Classification of the restricted simple Lie algebras,” Journal of Algebra,vol.114,no.1,pp.115– 259, 1988. [2] H. Strade, “The classification of the simple modular Lie algebras. IV. Determining the associated graded algebra,” Annals of Mathematics,vol.138,no.1,pp.1–59,1993. [3] V.G. Kac, “Lie superalgebras,” Advances in Mathematics,vol.26, no.1,pp.8–96,1977. [4] M. Scheunert, The theory of Lie superalgebras,vol.716ofLecture Notes in Mathematics, Springer, Berlin, Germany, 1979. [5] V. G. Kac, “Description of filtered Lie algebras with which gradedLiealgebrasofCartantypeareassociated,,”Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya,vol.8,no.4, article 801, 1974. [6] A. I. Kostrikin and I. R. Shafarevic, “Graded Lie algebras of finite characteristic,” Mathematics of the USSR-Izvestiya,vol.3,no.2, pp. 237–304, 1969. [7] N. Jin, “ad-nilpotent elements, quasi-nilpotent elements and invariant filtrations of infinite-dimensional Lie algebras of Cartan type,” Science in China,vol.35,no.10,pp.1191–1200,1992. [8] Y. Zhang and J. Nan, “Finite-dimensional Lie superalgebras 𝑊(𝑚, 𝑛, 𝑡) and 𝑆(𝑚, 𝑛,) 𝑡 of Cartan type,” Advances in Mathe- matics, vol. 27, no. 3, pp. 240–246, 1998. [9] Y. Zhang and H. Fu, “Finite-dimensional Hamiltonian Lie superalgebra,” Communications in Algebra,vol.30,no.6,pp. 2651–2673, 2002. [10] L. Ma, L. Chen, and Y. Zhang, “Finite-dimensional simple modular Lie superalgebra 𝑀,” Frontiers of Mathematics in China,vol.8,no.2,pp.411–441,2013. [11] X. Xu, L. Chen, and Y.Zhang, “On the modular Lie superalgebra Ω,” Journal of Pure and Applied Algebra,vol.215,no.5,pp.1093– 1101, 2011. [12] K. Zheng and Y. Zhang, “The natural filtration of finite dimensional modular Lie superalgebras of special type,” Abstract and Applied Analysis,vol.2013,ArticleID891241,7pages,2013. [13] V.G. Kac, “Classification of infinite-dimensional simple linearly compact Lie superalgebras,” Advances in Mathematics,vol.139, no.1,pp.1–55,1998. Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2014, Article ID 182354, 5 pages http://dx.doi.org/10.1155/2014/182354

Research Article Nonnegative Combined Matrices

Rafael Bru,1 Maria T. Gassó,1 Isabel Giménez,1 and Máximo Santana2

1 Institut de Matematica` Multidisciplinar, Universitat Politecnica` de Valencia,` 46022 Valencia,` Spain 2 Universidad Autonoma´ de Santo Domingo, Santo Domingo 10105, Dominican Republic

Correspondence should be addressed to Maria T. Gasso;´ [email protected]

Received 18 December 2013; Revised 11 March 2014; Accepted 15 March 2014; Published 7 April 2014

Academic Editor: Panayiotis Psarrakos

Copyright © 2014 Rafael Bru et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

−1 𝑇 The combined matrix of a nonsingular real matrix 𝐴 is the Hadamard (entrywise) product 𝐴∘(𝐴 ) .Itiswellknownthatrow (column) sums of combined matrices are constant and equal to one. Recently, some results on combined matrices of different classes of matrices have been done. In this work, we study some classes of matrices such that their combined matrices are nonnegative and obtain the relation with the sign pattern of 𝐴. In this case the combined matrix is doubly stochastic.

1. Introduction the combined matrix of different classes of matrices such as totally positive and totally negative matrices and also when 𝐴 Fiedler and Markham [1] studied matrices of the form 𝐶(𝐴) = −1 𝑇 is totally nonnegative and totally nonpositive. 𝐴∘(𝐴 ) ,thatis,thecombinedmatrixof𝐴,where𝐴 is a nonsingular matrix and ∘ means the Hadamard product. Combined matrices appear in the chemical literature where 2. Notation and Previous Results they represent the relative gain array (see [2]). Furthermore, the combined matrix gives the relation between the eigenval- Unless otherwise indicated, in this work all square matrices ues and diagonal entries of a diagonalizable matrix (see [3]). 𝐴 are 𝑛×𝑛nonsingular and real. Given an 𝑛×𝑛matrix Results for the combined matrix of a nonsingular matrix and 𝐴 𝑁 {1,...,𝑛} −1 ,wedenoteby the subset of indexes .Forthe also for the Hadamard product 𝐴∘𝐴 have been obtained, for subsets 𝛼, 𝛽 ⊆𝑁,thesubmatrixwithrowslyinginthesubset instance, in [1] where the behavior of the diagonal entries of 𝛼 andcolumnsinthesubset𝛽 is denoted by 𝐴[𝛼, 𝛽],andthe the combined matrix of a nonsingular matrix was completely principal submatrix 𝐴[𝛼, 𝛼] is abbreviated to 𝐴[𝛼].Similarly, describedandin[4] for the positive definite case. 𝐴(𝛼 | 𝛽) denotes the submatrix obtained from 𝐴 by deleting It is well known [3]thattherowandcolumnsumsofa rows lying in 𝛼 and columns in 𝛽,and𝐴(𝛼 | 𝛼) is abbreviated combined matrix are always equal to one. Then, if 𝐶(𝐴) is a to 𝐴(𝛼).Then,𝐴(𝑖 | 𝑗) denotes the submatrix obtained from nonnegative matrix, it has interesting properties and appli- 𝐴 by deleting row 𝑖 and column 𝑗,and𝐴[𝑖, 𝑗]𝑖𝑗 =𝑎 .Moreover, cations since it is a doubly stochastic matrix. For instance, 𝐴𝑖𝑗 denotes the (𝑖, 𝑗) minor; that is, 𝐴𝑖𝑗 = det 𝐴(𝑖 | 𝑗). in [5], there are two applications: the first one concerning a To avoid confusion with other matrices we will say topic in communication theory called satellite-switched and that 𝐴 is a nonnegative (positive) matrix if it is entrywise the second concerning a recent notion of doubly stochastic nonnegative (positive); that is, 𝑎𝑖𝑗 ≥0(𝑎𝑖𝑗 >0)forall𝑖, 𝑗 ∈𝑁, automorphism of a graph. Recently, in [6], some implications and we will denote it by 𝐴≥0(𝐴>0). Similar notation will on nonnegative matrices, doubly stochastic matrices, and be used for the nonpositive (negative) case; that is, 𝑎𝑖𝑗 ≤0 graph theory, namely, graph spectra and graph energy, are (𝑎𝑖𝑗 <0)forall𝑖, 𝑗 ∈𝑁, and we will denote it by 𝐴≤0 presented. (𝐴<0). Here,wefocusourworkonstudyingwhichmatriceshave Now we recall some classes of matrices that we are nonnegative combined matrices. More precisely, we study working on. 2 Journal of Applied Mathematics

Definition 1. An 𝑛×𝑛real matrix 𝐴=[𝑎𝑖𝑗 ] is said to have a Lemma 8. For any nonsingular matrix 𝐴,foranytwopermu- 𝑖+𝑗 𝑃 𝑄 𝑇 checkerboard pattern if sign(𝑎𝑖𝑗 )=(−1) or 𝑎𝑖𝑗 =0for all tation matrices and ,andforanytriangularmatrix one 𝑖, 𝑗 ∈𝑁.Whennoentryiszero,onewillsaythat𝐴 is strictly has checkerboard. (i) 𝐶(𝑃𝐴𝑄) = 𝑃𝐶(𝐴)𝑄, 𝐶(𝑃𝑇𝑄) = 𝑃𝐶(𝑇)𝑄 =𝑃𝑄 𝐶(𝑃𝑇𝑄) Definition 2. Amatrix𝐴 is said to be totally positive (nega- (ii) ;thatis, is a per- tive) if all its minors of any order are positive (negative). That mutation matrix, 𝑇 is, for every subset 𝛼, 𝛽 ⊆𝑁, det(𝐴[𝛼, 𝛽]) >0 (det(𝐴[𝛼, 𝛽]) < (iii) 𝐶(𝑃𝑇𝑃 )=𝐼. 0). It is denoted by TP (TN). From Definition 7 one can observe that, for 𝑛=2and Definition 3. Amatrix𝐴 is called totally nonpositive (non- 𝑛=3, 𝐶(𝐴) =𝐼 if and only if the matrix 𝐴 is permutationally negative) if all its minors of any order are nonpositive (non- similar to a triangular matrix. The natural question is whether negative). That is, for every subset 𝛼, 𝛽 ⊆𝑁, det(𝐴[𝛼, 𝛽]) ≤0 ornotthisisageneralequivalence.Asitissuggestedin[3] (det(𝐴[𝛼, 𝛽])). ≥0 It is denoted by TNP (TNN). (Problem 2, page 302) the equivalence is not true for some matrices with 𝑛≥4as the following example shows. Definition 4. An 𝑛×𝑛real matrix 𝐴 is said to be a 𝑃-matrix if, for every subset 𝛼⊆𝑁, det(𝐴[𝛼]). >0 Example 9. Consider the matrix

Definition 5. An 𝑛×𝑛real matrix 𝐴 is called an 𝑀-matrix if 𝑎00𝑎 [ ] 𝐴 can be written as 𝐴=𝑠𝐼−𝐵,with𝐵≥0and 𝑠≥𝜌(𝐵), 𝑏𝑐0𝑏 𝐴=[ ] 𝑎, 𝑏, 𝑐, 𝑑,𝑒,𝑓 =0.̸ (3) where 𝜌(𝐵) denotes the spectral radius of matrix 𝐵,thatis, [0𝑑𝑒0] the biggest absolute value of the eigenvalues of 𝐵. [0𝑑𝑒𝑓] For a matrix 𝐴, M(𝐴) denotes its comparison matrix; that 𝐶(𝐴) 𝐶(𝐴) =𝐼 is, It is easy to compute and to see that .However, 𝐴 is not permutationally similar to a triangular matrix since 󵄨 󵄨 𝐴 𝑚 =−󵄨𝑎 󵄨 ,𝑖=𝑗̸ is irreducible. M 𝐴 =[𝑚 ]={ 𝑖𝑗 󵄨 𝑖𝑗 󵄨 ( ) 𝑖𝑗 󵄨 󵄨 (1) 𝑚𝑖𝑖 = 󵄨𝑎𝑖𝑖󵄨 . Proposition 10. Let 𝐴 be a nonsingular matrix. If the combined matrix 𝐶(𝐴) is nonnegative and triangular, then 𝐶(𝐴) = Definition 6. An 𝑛×𝑛complex matrix 𝐴 is called an 𝐻-matrix 𝐼. if its comparison matrix M(𝐴) is an 𝑀-matrix. Proof. Recall that 𝐶(𝐴) is doubly stochastic since it is nonneg- Note that nonsingular 𝐻-matrices having singular com- ative. Suppose 𝐶(𝐴) is upper triangular. In this case 𝑐11 =1 parison matrix are included in this definition (see7 [ ]). istheonlypositiveentryofthefirstcolumnandthefirstrow. Remember that the Hadamard (or entrywise) product of Reasoning as before with the other columns of 𝐴 we conclude 𝐶(𝐴) =𝐼 two 𝑛×𝑛matrices 𝐴=[𝑎𝑖𝑗 ] and 𝐵=[𝑏𝑖𝑗 ] is the matrix 𝐴∘𝐵 = that . [𝑎𝑖𝑗 𝑏𝑖𝑗 ]. A simple case to determine matrices having their com- 𝐶(𝐴) ≥0 2×2 Definition 7. The combined matrix of a nonsingular real bined matrix is given for matrices in the −1 𝑇 following result. matrix 𝐴 is defined as 𝐶(𝐴) = 𝐴 ∘(𝐴 ) .Then,if𝐴=[𝑎𝑖𝑗 ],

Proposition 11. Let 𝐴=[𝑎𝑖𝑗 ] be a nonsingular 2×2matrix. −1 1 𝑖+𝑗 Then 𝐴 =[ (−1) 𝐴𝑗𝑖], det (𝐴) (2) 𝐶 (𝐴) ≥0⇐⇒𝑎11𝑎22𝑎12𝑎21 ≤0. (4) 1 𝑖+𝑗 𝐶 (𝐴) =[ (−1) 𝑎𝑖𝑗 𝐴𝑖𝑗 ]. 𝑎 𝑎 =0 𝐶(𝐴) =[ 01] det (𝐴) In particular, (i) if 11 22 ,then 10 ,(ii)if 𝑎12𝑎21 =0,then𝐶(𝐴) =𝐼, and finally (iii) if 𝑎11𝑎22𝑎12𝑎21 <0, then 𝐶(𝐴). >0 The elements of 𝐶(𝐴) will be denoted by 𝑐𝑖𝑗 .

It is clear that the combined matrix has the following Proof. Since 𝐶(𝐴𝑇)=𝐶(𝐴−1) 𝐶(𝐴) properties: (i) and (ii) is doubly sto- 𝑎11 𝑎12 −1 𝑇 1 𝑎22 −𝑎21 chastic if 𝐶(𝐴) is nonnegative. 𝐴=[ ], (𝐴 ) = [ ] (5) 𝑎21 𝑎22 det (𝐴) −𝑎12 𝑎11

3. Matrices with 𝐶(𝐴) ≥ 0 then 1 𝑎 𝑎 −𝑎 𝑎 𝐶 (𝐴) = [ 11 22 21 12] . Looking at the definition of a combined matrix it is easy to see (6) det (𝐴) −𝑎21𝑎12 𝑎11𝑎22 that the combined matrix of a triangular matrix is the identity matrix. Moreover, the following result from Lemma 5.4.2(𝑐) (⇒)If𝐶(𝐴),then ≥0 𝑎11𝑎22/ det(𝐴) ≥ 0 and −(𝑎21𝑎12/ (page 322)of[3]canbeestablished. det(𝐴)) ≥. 0 Therefore, (𝑎11𝑎22)(𝑎12𝑎21)≤0. Journal of Applied Mathematics 3

(⇐) Suppose first that det(𝐴) >.If 0 𝑎11𝑎22 <0,since It is well known that the combined matrix of an 𝑀- 𝑎11𝑎22𝑎12𝑎21 ≤0,then𝑎12𝑎21 ≥0, in which case det(𝐴) = matrix is an 𝑀-matrix (see [3]). The following result gives the 𝑎11𝑎22 −𝑎12𝑎21 <0.Then𝑎11𝑎22 ≥0and 𝑎12𝑎21 ≤0and hence equivalent conditions to have a nonnegative combined matrix 𝐶(𝐴). ≥0 Similarly, if det(𝐴) <,weobtain 0 𝑎11𝑎22 ≤0and of an 𝑀-matrix. 𝑎12𝑎21 ≥0and then 𝐶(𝐴). ≥0 Thetwofirstparticularcasescomefromthefactthat𝐶(𝐴) Theorem 17. Let A be an 𝑀-matrix. Then, the following con- is doubly stochastic, and the last case is because all entries are ditions are equivalent: different from zero. (i) 𝐶(𝐴), ≥0 Example 12. Consider (ii) 𝐶(𝐴) is diagonal, 3 2 (iii) 𝐶(𝐴) =𝐼, 11 [5 5] 𝐴=[ ]󳨐⇒𝐶(𝐴) = [ ] . 𝑎 𝐴 =1 𝑖∈𝑁 −2 3 [2 3] (7) (iv) 𝑖𝑖 𝑖𝑖 for all . −1 [5 5] Proof. Since 𝐴 is 𝑀-matrix, then 𝐴 is nonnegative and so The positivity of the matrix may play an important role in the off-diagonal elements of 𝐶(𝐴) are nonpositive and the our case. According to the definition of the combined matrix, diagonal elements are positive. Then, 𝐶(𝐴) ≥0 if and only we have the following. if 𝐶(𝐴) is diagonal. The last two equivalences follow because the combined matrix is doubly stochastic. Theorem 13. Let 𝐴≥0(𝐴≤0). Then 𝐶(𝐴) ≥0 ifandonlyif −1 −1 𝐴 ≥0(𝐴 ≤0). Despite the fact that the nonnegativity of the combined matrix of an 𝑀-matrix is reduced to the identity matrix, it Matrices with the property described in Theorem 13 is easy to find 𝐻-matrices for which its combined matrix are necessarily nonnegative monomial matrices, as are the is nonnegative and different from the identity matrix as the corresponding combined matrices. Monomial matrices are following example shows. indeed permutationally similar to diagonal matrices and thus are orthogonal matrices. Below we prove a related result about Example 18. The nonsingular matrices a type of orthogonal matrices that was recently introduced in [8]. 41−2 41−2 [ ] [ ] 𝐴1 = −1 5 −3 ,𝐴2 = −1 5 3 , 𝐴 𝐺 Definition 14. Anonsingularmatrix is called a -matrix if [ 227] [ 2−27] two nonsingular diagonal matrices 𝐷1 and 𝐷2 exist such that (12) (𝐴−1)𝑇 =𝐷𝐴𝐷 31−2 1 2. [ ] 𝐴3 = −1 4 3 , 5−27 Theorem 15. Let 𝐴 be a nonsingular 𝐺-matrix such that 𝐷1 ≥ [ ] 0 and 𝐷2 ≥0.Then,thecombinedmatrixof𝐴 is nonnegative. are 𝐻-matrices and each one has positive combined matrix: Proof. In this case, we have 164 1 24 𝑇 1 𝐶 (𝐴) =𝐴∘(𝐴−1) =𝐴∘(𝐷𝐴𝐷 )=𝐷 (𝐴∘𝐴) 𝐷 (8) 𝐶(𝐴 )= [ 11 160 18 ] , 1 2 1 2 1 189 [ 14 28 147] which is nonnegative. 164 13 16 Example 16. The matrix 1 [ ] 𝐶(𝐴2)= 3 160 30 , (13) 21 193 𝐴=[ ] [ 26 20 147] 1−1 (9) is a 𝐺-matrix, since 102 22 36 1 [ ] 1 1 𝐶(𝐴3)= 3 124 33 . 160 55 14 91 𝑇 [3 3 ] [ ] (𝐴−1) = [ ] [1 2] (10) − It should be noted that 𝐴3 is nonsingular but M(𝐴3) is a [3 3] singular 𝑀-matrix. is the result of 𝐷𝐴𝐷,where𝐷=diag(1/√6, √2/3).Then,the Now, we study the positivity of the combined matrix of combined matrix is 2 1 totally positive and totally negative matrices. [ ] [3 3] Theorem 19. If 𝐴 is a TP-matrix, then 𝐶(𝐴) is strictly 𝐶 (𝐴) = [ ] ≥0. (11) 1 2 checkerboard. In addition, if 𝐴 is a TN-matrix, then −𝐶(𝐴) is [3 3] strictly checkerboard. 4 Journal of Applied Mathematics

Proof. It is straightforward noting that 𝐶(𝐴) is checkerboard Proof. (ii)→(i).If𝐶(𝐴) =𝐼,itisobviousthat𝐶(𝐴) is in both cases. nonnegative. (i)→(ii).Letussupposethat𝐶(𝐴) is nonnegative. We Then, the combined matrix of a TP-matrix or a TN- note that 𝑎𝑖𝑗 𝐴𝑖𝑗 =0whenever 𝑖+𝑗 = 2𝑘+1, 𝑘∈N, matrix is not nonnegative. However, we have the following by Proposition 23. Then it remains to prove that 𝑎𝑖𝑗 𝐴𝑖𝑗 =0 result. whenever 𝑖+𝑗 = 2𝑘and 𝑖 =𝑗̸ . We work by contradiction. For this, we suppose that there exists 𝑎𝑖𝑗 𝐴𝑖𝑗 =0̸ with 𝑖+𝑗 = 2𝑘 Theorem 20. If 𝐴 is a TP-matrix (TN-matrix), then 𝑆𝐶(𝐴)𝑆 > 𝑇 𝑛−1 𝑖<𝑗 𝑆(𝐴−1) 𝑆 0 (−𝑆𝐶(𝐴)𝑆), >0 where 𝑆=diag(1,−1,1,−1,...,(−1) ). and .ByProposition 24 is a nonsingular TNN- −1 𝑇 matrix. Applying Proposition 22 to both 𝐴 and 𝑆(𝐴 ) 𝑆 we To study the combined matrix of a TNN-matrix, we need have that 𝑎𝑖,𝑗−1 =0̸ and 𝐴𝑖,𝑗−1 =0̸ . Therefore, 𝑎𝑖,𝑗−1𝐴𝑖,𝑗−1 =0̸ , some auxiliary results. with 𝑖+(𝑗−1)= 2𝑘−1,andthiscontradictstheresultof Proposition 23.Thus,when𝑖<𝑗,wehavethat𝑎𝑖𝑗 𝐴𝑖𝑗 =0. Proposition 21 𝐴 (see [9], Corollary 3.8). If is a nonsingular Then, 𝐶(𝐴) is a lower triangular and nonnegative matrix. 𝐴 TNN-matrix, then is a P-matrix. Hence, 𝐶(𝐴) =𝐼 by Proposition 10. Proposition 22. 𝐴 𝑎 =0̸ If is a nonsingular TNN-matrix and 𝑖𝑗 Now, let us figure out the combined matrix of a TNP- 𝑗>𝑖 𝑎 =0̸ for some ,then 𝑖,𝑗−1 . matrix. Since all minors of TNP-matrices are nonpositive, it is clear that the combined matrix of a nonsingular TNP-matrix Proof. If 𝑗=𝑖+1,then𝑎𝑖,𝑗−1 =𝑎𝑖𝑖, which is positive since 𝐴 𝑃 𝑗≥𝑖+2 is not, in general, nonnegative. In fact, we have this simple is a -matrix by Proposition 21.For , consider the result. submatrix 𝑎 𝑎 Theorem 26. Let 𝐴 be a nonsingular TNP-matrix. Then 𝐶(𝐴) [ 𝑖,𝑗−1 𝑖𝑗 ] 𝑎 𝐴 =0 𝑖+𝑗=2𝑘 𝑎 𝑎 (14) is nonnegative if and only if 𝑖𝑗 𝑖𝑗 ,whenever , 𝑗−1,𝑗−1 𝑗−1,𝑗 𝑘∈N. and let us work by contradiction. Suppose that 𝑎𝑖,𝑗−1 =0. However, there exist TNP-matrices with nonnegative Since 𝐴 is a nonsingular TNN-matrix, then 𝑎𝑗−1,𝑗−1 >0and combined matrices as the following example shows. 𝑎𝑖𝑗 >0.Then Example 27. Consider the totally nonpositive matrix 0𝑎𝑖𝑗 det [ ]<0, (15) 𝑎 𝑎 0−1 𝑗−1,𝑗−1 𝑗−1,𝑗 𝐴=[ ]. −1 0 (17) which is a contradiction. The combined matrix is Proposition 23. If 𝐴 is a nonsingular TNN-matrix, then the 01 following conditions are equivalent: 𝐶 (𝐴) =[ ], 10 (18) (i) 𝐶(𝐴), ≥0

(ii) 𝑎𝑖𝑗 𝐴𝑖𝑗 =0, 𝑖+𝑗=2𝑘+1, 𝑘∈N. which is nonnegative.

Proof. Since 𝐴 is TNN, 𝑎𝑖𝑗 ≥0and 𝐴𝑖𝑗 ≥0for all 𝑖,.Then 𝑗 The question now is to know whether or not there are more TNP-matrices with nonnegative combined matrices. As 2×2 1 𝑖+𝑗 ≥0 if 𝑖+𝑗=2𝑘 we see below, only TNP-matrices of size may have this 𝑐𝑖𝑗 = (−1) 𝑎𝑖𝑗 𝐴𝑖𝑗 { (16) det (𝐴) ≤0 if 𝑖+𝑗=2𝑘+1. property. To prove this we need some auxiliary results concerning TNP-matrices.

Therefore, 𝐶(𝐴) is nonnegative if and only if 𝑎𝑖𝑗 𝐴𝑖𝑗 =0,when Proposition 28. Let 𝐴 be an 𝑛×𝑛nonsingular TNP-matrix 𝑖+𝑗=2𝑘+1. with 𝑛≥3. If there exists an index 𝑗≥2such that 𝐴1𝑗 =0̸ , 𝐴 =0̸ Proposition 24 (see [9], Theorem 3.3). If 𝐴 is a nonsingular then 1,𝑗−1 . −1 TNN-matrix, then 𝑆𝐴 𝑆 is also a nonsingular TNN-matrix, 𝑛−1 Proof. Suppose by contradiction that there is an 𝐴1𝑗 =0̸ and where 𝑆=diag(1,−1,...,(−1) ). −1 𝐴1,𝑗−1 =0for some 𝑗≥2.Since𝐴 is nonsingular, there 𝑡>1 𝐴 =0̸ Thenwecanestablishthefollowingresult. exists an index such that 𝑡,𝑗−1 and then

Theorem 25. Let 𝐴 be a nonsingular TNN-matrix. Then the 𝐴1,𝑗−1 𝐴1𝑗 det [ ]<0, (19) following conditions are equivalent: 𝐴𝑡,𝑗−1 𝐴𝑡𝑗 𝐶(𝐴) (i) is nonnegative, which is a contradiction with the signature (1,1,...,1,−1)of −1 (ii) 𝐶(𝐴) =𝐼. the matrix 𝑆𝐴 𝑆 (see [10]). Journal of Applied Mathematics 5

Corollary 29. Let 𝐴 be an 𝑛×𝑛nonsingular TNP-matrix with Proof. The computation of the combined matrix gives 𝑛≥3. 1 𝑎𝑑 −𝑐𝑏 𝐴 =0̸ 𝑗≥2 𝐴 =0̸ 𝑡≤𝑗 𝐶 (𝐴) = [ ]. (22) (i) If 1𝑗 for some ,then 1𝑡 for all . det 𝐴 −𝑐𝑏 𝑎𝑑 (ii) If 𝑐1𝑗 =0̸ for some 𝑗≥2,then𝑐1𝑡 =0̸ , 2≤𝑡≤𝑗. Then, since det 𝐴<0, 𝐶(𝐴) ≥0 if and only if 𝑎𝑑.Inthis =0 𝐴=−𝑏𝑐 𝐶(𝐴) =[ 01] Proof. (i) The proof follows from Proposition 28. case, det and 10 . (ii) Since 𝐴 is TNP, 𝑎1𝑡 =0̸ for all 𝑡≥2(see [11], Theorem In brief, only 2×2antitriangular TNP-matrices have their 2.1 (i)). Then, using part (i) of this corollary, we conclude that combined matrix nonnegative. 𝑐1𝑡 =𝑎1𝑡𝐴1𝑡 =0̸ whenever 2≤𝑡≤𝑗.

In the following theorem we show how the first row and Conflict of Interests thesecondcolumnofanonnegativecombinedmatrixofa TNP-matrix are. The authors declare that there is no conflict of interests regarding the publication of this paper. Theorem 30. Let 𝐴 be an 𝑛×𝑛nonsingular TNP-matrix with 𝑛≥3.If𝐶(𝐴) is nonnegative, then the first row of 𝐶(𝐴) is 𝑇 Acknowledgments (0,1,0,...,0)and the second column is (1,0,...,0) . The authors would like to thank the referees for their Proof. Again, since 𝐴 is TNP, 𝑎1𝑡 =0̸ for all 𝑡≥2.Further- suggestions that have improved the reading of this paper. This more, 𝐴13 =𝐴15 =⋅⋅⋅=𝐴1,2𝑘+1 =0, 𝑘∈N, 𝑘≤(𝑛−1)/2by research is supported by Spanish DGI (Grant no. MTM2010- Theorem 26 and 𝐴1𝑗 =0for 𝑗=3,4,...,𝑛by Proposition 28. 18674). Then, the proof follows since 𝐶(𝐴) is doubly stochastic and 𝑐 =0 11 . References

Now, we can give the main theorem on combined matri- [1] M. Fiedler and T. L. Markham, “Combined matrices in special ces of TNP-matrices; that is, we are going to prove that there classes of matrices,” Linear Algebra and Its Applications,vol.435, does not exist any nonsingular TNP-matrix 𝐴 of size 𝑛≥3 no. 8, pp. 1945–1955, 2011. such that its combined matrix is nonnegative. [2]T.J.McAvoy,Interaction Analysis: Principles and Applications, vol.6ofMonograph Series, Instrument Society of America, 1983. Theorem 31. Let 𝐴 be an 𝑛×𝑛nonsingular TNP-matrix with 𝑛≥3 𝐶(𝐴) [3] R. A. Horn and C. R. Johnson, Topics in Matrix Analysis,Cam- .Then is not nonnegative. bridge University Press, Cambridge, UK, 1991. Proof. Let us work by contradiction and suppose that there [4] M. Fiedler, “Relations between the diagonal elements of two 𝐴 𝑘×𝑘 𝑘≥ mutually inverse positive definite matrices,” Czechoslovak Math- exists a nonsingular TNP-matrix of size ,with ematical Journal,vol.14,no.89,pp.39–51,1964. 3 and 𝐶(𝐴).By ≥0 Theorem 30 we know the structure of 𝐶(𝐴) [5] R. A. Brualdi, “Some applications of doubly stochastic matrices,” thefirstrowandthesecondcolumnof .Letusfocuson Linear Algebra and Its Applications, vol. 107, pp. 77–100, 1988. its first column where its first entry is 𝑐11 =0.Since𝐶(𝐴) is [6] B. Mourad, “Generalization of some results concerning eigen- nonnegative and so doubly stochastic, then there must exist values of a certain class of matrices and some applications,” an index 𝑡>1such that 𝑐𝑡1 =0̸ ,andthen𝐴𝑡1 =0̸ .Further,by 𝐴 =0 𝐴 =0̸ Linear and Multilinear Algebra,vol.61,no.9,pp.1234–1243, Theorem 30 and Theorem 2.1 (i) of [11] 𝑡2 and 12 2013. by Theorem 30.Then [7]R.Bru,C.Corral,I.Gimenez,´ and J. Mas, “Classes of general 𝐻 𝐴 𝐴 -matrices,” Linear Algebra and Its Applications,vol.429,no. [ 11 12]<0. 10, pp. 2358–2366, 2008. det 𝐴 𝐴 (20) 𝑡1 𝑡2 [8] M. Fiedler and F. J. Hall, “G-matrices,” Linear Algebra and Its Applications,vol.436,no.3,pp.731–741,2012. 𝑆𝐴−1𝑆 This is a contradiction with signature of according to [9] T. Ando, “Totally positive matrices,” Linear Algebra and Its [10]. Applications,vol.90,pp.165–219,1987. [10] S. Fallat and P.van den Driessche, “On matrices with all minors Finally,weprovewhatkindoftotallynonpositivematri- 𝑛=2 negative,” The Electronic Journal of Linear Algebra,vol.7,pp.92– ces of size havetheircombinedmatrixnonnegativein 99, 2000. the following result. [11] J. M. Pena,˜ “On nonsingular sign regular matrices,” Linear Alge- bra and Its Applications,vol.359,no.1–3,pp.91–100,2003. Theorem 32. Let the 2×2TNP-matrix be −𝑎 −𝑏 𝐴=[ ], −𝑐 −𝑑 (21) where 𝑎, 𝑏, 𝑐, 𝑑 are nonnegative. Then 𝐶(𝐴) ≥0 ifandonlyifat 01 leastoneoftheentries𝑎 or 𝑑 is zero. In this case, 𝐶(𝐴) =[ 10].