Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2014, Article ID 795203, 5 pages http://dx.doi.org/10.1155/2014/795203

Research Article An Algebraic Relation between Consimilarity and Similarity of Quaternion Matrices and Applications

Tongsong Jiang,1,2 Xuehan Cheng,3 and Sitao Ling4

1 College of Science, Linyi University, Linyi, Shandong 276005, China 2 College of Mathematics and System Science, Shandong University of Science and Technology, Qingdao, Shandong 266590, China 3 College of Mathematics and Statistics Science, Ludong University, Yantai, Shandong 264025, China 4 State Key Laboratory for Geomechanics and Deep Underground Engineering, Department of Mathematics, China University of Mining and Technology, Xuzhou, Jiangsu 221116, China

Correspondence should be addressed to Sitao Ling; [email protected]

Received 12 May 2014; Revised 15 August 2014; Accepted 21 August 2014; Published 1 September 2014

Academic Editor: Fan Min

Copyright © 2014 Tongsong Jiang 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.

This paper, by means of complex representation of a quaternion matrix, discusses the consimilarity of quaternion matrices, and obtains a relation between consimilarity and similarity of quaternion matrices. It sets up an algebraic bridge between consimilarity and similarity, and turns the theory of consimilarity of quaternion matrices into that of ordinary similarity of complex matrices. This paper also gives algebraic methods for finding coneigenvalues and coneigenvectors of quaternion matrices by means of complex representation of a quaternion matrix.

1. Introduction If 𝐴 and 𝐵 are both quaternion matrices of 𝑛×𝑛,theyaresaid −1 to be consimilar if 𝑆 𝐴𝑆=𝐵̃ holds for some 𝑛×𝑛 nonsingular An antilinear operator 𝑇 is a mapping from one complex 𝑠 𝑐𝑠 quaternion matrix 𝑆.Write𝐴 ∼𝐵if 𝐴 is similar to 𝐵, 𝐴 ∼𝐵 vector space 𝑉 into another 𝑊, which is additive and conju- 𝑝𝑠 gate homogeneous, that is, for all 𝛼, 𝛽 ∈𝑉 and any complex if 𝐴 is consimilar to 𝐵,and𝐴 ∼𝐵if 𝐴 is permutation sim- number 𝑎, 𝑇(𝛼 + 𝛽) = 𝑇(𝛼), +𝑇(𝛽) 𝑇(𝑎𝛼) = 𝑎𝑇(𝛼) in which ilar to 𝐵. Permutation similarity is both a similarity and con- 𝑎 is the conjugate of 𝑎.Two𝑛×𝑛complex matrices 𝐴, 𝐵 are similarity relations. A quaternion 𝜆 is said to be the right −1 ̃ said to be consimilar if 𝑆 𝐴𝑆=𝐵for some 𝑛×𝑛nonsingular coneigenvalueattributedto𝐴 if the con-eigenequation 𝐴𝛼= complex matrix 𝑆. Consimilarity of complex matrices arises 𝛼𝜆 holds. The consimilarity of quaternion matrices is natural as a result of studying an antilinear operator referred to dif- extension of that of complex matrices and will have potential ferent bases in complex vector spaces, and the theory of applications in the study of theory and numerical computa- consimilarity of complex matrices plays an important role tionsinmodernquantummechanics,andsoforth. in quantum mechanics [1, 2]. Complex consimilarity is an Let R be the real number field, C = R ⊕ R𝑖 the complex equivalent relation and has been studied [2–5]. number field, and H = R ⊕ R𝑖⊕R𝑗⊕R𝑘 the quaternion 2 2 2 𝑚×𝑛 In recent years, applications of quaternion matrices are field, where 𝑖𝑗=−𝑗𝑖=𝑘, 𝑖 =𝑗 =𝑘 =−1. F denotes 𝑚×𝑛 getting more and more important and extensive in quantum the set of all 𝑚×𝑛matrices on a field F.For𝐴∈H ,let 𝑇 mechanics, rigid mechanics, and control theory [6–17]; it is 𝐴 be the transpose of 𝐴 and 𝐴 the conjugate matrix of 𝐴. becoming more and more necessary to study the theory and For a quaternion 𝛼=𝑎1 +𝑎2𝑖+𝑎3𝑗+𝑎4𝑘 with 𝑎𝑡 ∈ R,we methods of quaternion matrices. In paper [18], the author use 𝛼=𝑎1 −𝑎2𝑖−𝑎3𝑗−𝑎4𝑘 and 𝛼=𝑎̃ 1 −𝑎2𝑖+𝑎3𝑗−𝑎4𝑘 introduced concepts of consimilarity of quaternion matrices. todenotetheconjugateand𝑗-conjugate of 𝛼,respectively. 2 Journal of Applied Mathematics

𝑚×𝑛 𝑛×𝑛 Let 𝐴=𝐴1 +𝐴2𝑖+𝐴3𝑗+𝐴4𝑘,where𝐴𝑠 ∈ R ; define Proposition 1. Let 𝐴∈H .Then 𝐴=−𝑗𝐴𝑗=𝐴̃ −𝐴 𝑖+𝐴 𝑗−𝐴 𝑘 𝑗 𝐴 1 2 3 4 to be -conjugate of .It 𝐴𝜎 ̃ ̃ (1) the real eigenvalues of complex representation is easy to verify that 𝑗𝐴𝑗 =−𝐴, 𝐴=𝐴,and appear in pairs, and the imaginary eigenvalues of com- 𝜎 ̃ ̃ ̃ ̃ ̃̃ plex representation 𝐴 appear in conjugate pairs; that 𝐴+𝐵= 𝐴 + 𝐵, 𝐴𝐶 = 𝐴𝐶, (1) 𝜎 is, if complex eigenvalue 𝜆 is an eigenvalue of 𝐴 ,then 𝜎 for any two 𝑛×𝑛matrices 𝐴, 𝐵 and a 𝑛×𝑛matrix 𝐶 with complex eigenvalue 𝜆 is also an eigenvalue of 𝐴 , quaternion entries. Obviously, 𝐴=̃ 𝐴 if 𝐴 is a complex matrix. (2) the real Jordan blocks 𝐽𝑟(𝜆) of complex representation By means of real representation of complex matrices 𝜎 𝐴 appear in pairs, and the imaginary Jordan blocks in paper [5], we studied the properties of consimilarity of 𝐽 (𝜆) 𝐴𝜎 complex matrices and gave a relation between consimilarity 𝑟 of complex representation appear in conjugate pairs; that is, if 𝐽𝑟(𝜆) is a Jordan block of complex and similarity of complex matrices. This paper, by means of 𝐴𝜎 𝜆 a complex representation of a quaternion matrix, studies the representation related to complex eigenvalue ,then 𝐽 (𝜆) 𝐴𝜎 relation between consimilarity and similarity of quaternion 𝑟 is a Jordan block of complex representation matrices and derives an algebraic relation between consim- related to complex eigenvalue 𝜆. ilarity and similarity on quaternion field. Finally this paper gives an application on coneigenvalues and coneigenvectors The following Proposition 2 comes from the fact that ̃ ̃ of quaternion matrices. 𝑗𝐴𝑗 =−𝐴, (𝐽𝑟(𝑎 + 𝑏𝑖))𝑗(𝑖 + 𝑗) = (𝑖𝑟 +𝑗)𝐽 (−𝑏 + 𝑎𝑗) and ((𝐴1 + ̃ 𝐴2𝑖)𝑗)(𝑖 + 𝑗) = (𝑖 + 𝑗)(−𝐴2 +𝐴1𝑗) by direct calculation for 2. Preliminaries real numbers 𝑎 and 𝑏,andrealmatrices𝐴1 and 𝐴2. 𝐴 𝑛×𝑚 For a quaternion matrix of dimension , denote by Proposition 2. Let 𝐴 be a quaternion matrix. Then 𝐴=𝐵1 +𝐵2𝑖+𝐵3𝑗+𝐵4𝑘≡𝐴1 +𝐴2𝑗,where𝐴1 =𝐵1 +𝐵2𝑖, ̃ ̃ 𝐴2 =𝐵3 +𝐵4𝑖. The complex representation of the quaternion (1) 𝐴𝑗 =𝑗𝐴;thismeansthat𝐴 is consimilar to 𝐴;thatis, 𝐴 𝑐𝑠 matrix is defined [19]tobe 𝐴 ∼ 𝐴̃; 𝐴 −𝐴 𝑐𝑠 𝜎 2 1 2𝑛×2𝑚 𝐽=𝐽(𝑎 + 𝑏𝑖) 𝑎, 𝑏 ∈ R 𝐽𝑗 ∼ 𝐴 =[ ]∈C ; (2) (2) if 𝑟 is a Jordan block, ,then 𝐴1 𝐴2 𝐽𝑟(−𝑏 + 𝑎𝑗); 𝜎 𝑛×𝑛 the complex matrix 𝐴 is known as complex representation of (3) if 𝐴=𝐴1 +𝐴2𝑖 is a complex matrix, 𝐴1,𝐴2 ∈ R , 𝜎 the quaternion matrix 𝐴.Itiseasytoverifythat𝐴𝜎 =(𝐴)̃ . then 𝑛×𝑚 𝑚×𝑝 Let 𝐴, 𝐵 ∈ H , 𝐶∈H , 𝑎∈R. Then by the definition 𝑐𝑠 𝑐𝑠 𝐴𝑗 ∼−𝐴2 +𝐴1𝑗, 𝐴𝑗 ∼(𝐴1 +𝐴2𝑗) 𝑗. (8) ofcomplexrepresentationweeasilygetthefollowingresults: 𝜎 𝜎 𝜎 𝜎 𝜎 (𝐴+𝐵) =𝐴 +𝐵 , (𝑎𝐴) =𝑎𝐴 ; (3) Based on the quaternion field H,theauthorin[18]gave the following results. 𝑇 𝜎 𝜎 𝑄 𝐴 𝑄 = 𝐴𝜎 =(𝐴)̃ ; (4) 𝑛 𝑚 𝑛×𝑛 Proposition 3 (see [18]). If 𝐴, 𝐵 ∈ H ,then 𝜎 𝜎 𝜎 𝜎 ̃ 𝜎 ̃ 𝜎 𝜎 (𝐴𝐶) =𝐴 𝑄𝑚𝐶 =𝐴 (𝐶) 𝑄𝑝 =𝑄𝑛(𝐴) 𝐶 , (5) 𝑐𝑠 𝑠 𝑠 𝑠 𝐴 ∼ 𝐵⇐⇒𝑗𝐴∼ 𝑗𝐵 ⇐⇒ 𝐴𝑗 ∼ 𝐵𝑗 ⇐⇒ 𝑗𝐴 ∼ 𝐵𝑗. (9) 0𝐼𝑡 in which 𝑄𝑡 =[ ] with 𝐼𝑡 being the identity matrix, −𝐼𝑡 0 𝑛×𝑛 2 𝑇 𝑇 Proposition 4 (see [18]). If 𝐴∈H ,then and 𝑄𝑡 =−𝐼2𝑡, 𝑄𝑡𝑄𝑡 =𝑄𝑡 𝑄𝑡 =𝐼2𝑡.Clearly,from(5)we know that a quaternion matrix 𝐴 is nonsingular if and only if 𝑐𝑠 𝑐 𝜎 𝐴 ∼𝐽𝑛 (𝜆1)⊕𝐽𝑛 (𝜆2)⊕⋅⋅⋅⊕𝐽𝑛 (𝜆𝑟)≡𝐽 , (10) complex representation matrix 𝐴 is nonsingular. 1 2 𝑟 𝐴 𝑛×𝑛 2𝑛×1 𝜎 For 𝐴∈H and 𝛼∈C ,if𝐴 𝛼=𝜆𝛼,thenby(4)we in which 𝜆𝑡 =𝑎𝑡 +𝑏𝑡𝑗 are right coneigenvalues of 𝐴, 𝑎𝑡,𝑏𝑡 are 𝑐 have real numbers, and 𝑎𝑡 ≥0. 𝐽𝐴 is uniquely determined by 𝐴 up 𝜎 𝑇 𝑇 𝜎 𝑇 𝑇 𝑐 to the order of Jordan blocks 𝐽𝑛 (𝜆𝑡),and𝐽𝐴 is said to be the 𝐴 (𝑄𝑛 𝛼) =𝜆(𝑄𝑛 𝛼) ⇐⇒ 𝐴 (𝑄𝑛 𝛼) = 𝜆(𝑄𝑛 𝛼) . (6) 𝑡 𝐴 𝜎 Jordan canonical form of under consimilarity. This means that the eigenvalues of complex representation 𝐴 appear in conjugate pairs. 3. An Algebraic Relation between In the same manner, for a Jordan block 𝐽𝑟(𝜆) of an 𝜎 eigenvalue 𝜆,if𝐴 𝑋=𝑋𝐽𝑟(𝜆),thenby(4)wehave Consimilarity and Similarity 𝜎 𝑇 𝑇 𝜎 𝑇 𝑇 𝐴 (𝑄𝑛 𝑋) = (𝑄𝑛 𝑋)𝑟 𝐽 (𝜆) ⇐⇒ 𝐴 (𝑄𝑛 𝑋) = (𝑄𝑛 𝑋)𝑟 𝐽 (𝜆) . This section gives an algebraic relation between consimilarity and similarity of quaternion matrices by means of the com- (7) plex representation in (2). 𝐴, 𝐵 ∈ H𝑛×𝑛 𝐴 And this means Jordan blocks 𝐽𝑟(𝜆) of complex representa- Let be two quaternion matrices. If is 𝜎 tion 𝐴 appear in conjugate pairs. consimilar to 𝐵, then there exists a nonsingular quaternion ̃ 𝜎 𝜎 𝜎 𝜎 From the statement above we have the following result. matrix 𝑆 such that 𝐴𝑆 = 𝑆𝐵,by(5) 𝐴 𝑄𝑛𝑆 =𝑄𝑛𝑆 𝐵 . Journal of Applied Mathematics 3

𝜎 This means that if 𝐴 is consimilar to 𝐵,then𝐴 is similar to The proof of the Theorem 5 is constructive, and the 𝜎 𝐵 . following results come from the proof above. 𝐴𝜎 𝐵𝜎 𝐴𝜎 𝐵𝜎 Conversely, if is similar to ,then and have the Corollary 7. 𝐴∈H𝑛×𝑛 𝜆 , 𝜆 ,...,𝜆 , 𝜆 Let . Then the following statements are same eigenvalues. By Proposition 1,let 1 1 𝑟 𝑟 be equivalent. all eigenvalues and 𝐽𝑛 (𝜆1), 𝐽𝑛 (𝜆1),...,𝐽𝑛 (𝜆𝑟), 𝐽𝑛 (𝜆𝑟) corre- 1 1 𝑟 𝑟 𝐴𝜎 sponding Jordan blocks. There exists a complex and full-rank (1) The Jordan canonical form of complex matrix under 𝑋 𝐴𝜎𝑋 =𝑋𝐽 (𝜆 ) similarity is matrix 𝑗 by [20,chapter6.7]suchthat 𝑡 𝑡 𝑛𝑡 𝑡 , 𝑡=1,...,𝑟.Thenby(4), we have 𝐽=∑ ⊕𝐽𝑛 (𝜆𝑡)⊕∑ ⊕𝐽𝑛 (𝜆𝑡), 𝑡 𝑡 (16) 𝑡 𝑡 𝐴𝜎𝑋 =𝑋𝐽 (𝜆 ), 𝐴𝜎 (𝑄𝑇𝑋 )=(𝑄𝑇𝑋 )𝐽 (𝜆 ), 𝑡 𝑡 𝑛𝑡 𝑡 𝑛 𝑡 𝑛 𝑡 𝑛𝑡 𝑡 where 𝜆𝑡 =𝑎𝑡 +𝑏𝑡𝑖, 𝑎𝑡,𝑏𝑡 ∈ R, 𝑏𝑡 ≥0, 𝑡 = 1,...,𝑟,are (11) 𝜎 eigenvalues of 𝐴 . and (11)isequivalentto (2) The Jordan canonical form of quaternion matrix 𝐴 𝐽 (𝜆 )0 under consimilarity is 𝜎 𝑇 𝑇 𝑛𝑡 𝑡 𝐴 (𝑋𝑡,𝑄𝑛 𝑋𝑡)=(𝑋𝑡,𝑄𝑛 𝑋𝑡)[ ]. (12) 𝑐 0𝐽𝑛 (𝜆𝑡) 𝐽 = ∑ ⊕𝐽𝑛 (𝜇𝑡), 𝑡 𝑡 (17) 𝑡 Therefore there exists a nonsingular complex matrix 𝑇= where 𝜇𝑡 =𝑏𝑡 +𝑎𝑡𝑗, 𝑎𝑡,𝑏𝑡 ∈ R, 𝑏𝑡 ≥0, 𝑡 = 1,...,𝑟,are (𝑍,𝑇 𝑄 𝑍) ∈ C2𝑛×2𝑛 𝑍∈C2𝑛×𝑛 𝑛 , such that coneigenvalues of 𝐴.

𝑛×𝑛 ∑ ⊕𝐽𝑛 (𝜆𝑡)0 Corollary 8. 𝐴∈H 𝜎 [ 𝑡 ] Let .Then 𝑇−1𝐴𝜎𝑇 = (𝐽𝑗) = [ 𝑡 ] , (13) 𝑐𝑠 0 ∑ ⊕𝐽𝑛 (𝜆𝑡) 𝑐 𝑡 𝐴 ∼𝐽 (𝜇 )⊕𝐽 (𝜇 )⋅⋅⋅⊕𝐽 (𝜇 )≡𝐽, (18) [ 𝑡 ] 𝑛1 1 𝑛2 2 𝑛𝑟 𝑟 𝜇 =𝑏+𝑎𝑗 = (1/2𝑖)(𝜆 − 𝜆 ) + (1/2)(𝜆 + 𝜆 )𝑗 in which 𝐽=∑𝑡 ⊕𝐽𝑛 (𝜆𝑡), 𝜆𝑡 =𝑎𝑡 +𝑏𝑡𝑖, 𝑎𝑡,𝑏𝑡 are real numbers, where 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 are 𝑡 𝐴 𝜆 =𝑎+𝑏 𝑖 𝑎 ,𝑏 ∈ R 𝑏 ≥0 𝑡=1,...,𝑟 and 𝑏𝑡 ≥0. coneigenvalues of , 𝑡 𝑡 𝑡 , 𝑡 𝑡 , 𝑡 , , 𝑍 1 𝑛×𝑛 𝜆 , 𝜆 ,...,𝜆 , 𝜆 Let 𝑍=[𝑍 ], 𝑍1,𝑍2 ∈ C ,and𝑆=𝑍2 +𝑍1𝑗.Then in which 1 1 𝑟 𝑟 are all eigenvalues of complex rep- 2 𝜎 𝑍1 −𝑍2 𝜎 𝐴 𝐽 (𝜆 ), 𝐽 (𝜆 ),...,𝐽 (𝜆 ), 𝐽 (𝜆 ) 𝑇=[ ]=𝑆 𝑇 resentation ,and 𝑛1 1 𝑛1 1 𝑛𝑟 𝑟 𝑛𝑟 𝑟 corre- 𝑍 𝑍 . From the nonsingular complex matrix , 𝑐 2 1 sponding Jordan blocks in (16). 𝐽 is uniquely determined by 𝐴 we get 𝑆 is a nonsingular quaternion matrix; by (1)and(5), 𝐽 (𝜇 ) 𝐽𝑐 up to the order of Jordan blocks 𝑛𝑡 𝑡 ,and is said to be the (13)isequivalentto Jordan canonical form of 𝐴 under consimilarity. 𝜎 𝜎 𝜎 𝜎 𝜎 𝜎 ̃ ̃ 𝑛×𝑛 𝐴 𝑆 =𝑆 (𝐽𝑗) ⇐⇒ ( 𝐴 𝑆) =(𝑆(𝐽𝑗)) Corollary 9. Let 𝐴∈H .Then𝐴 has at least a coneigen- (14) value 𝜇𝑡 = Re(𝜇𝑡)+Im(𝜇𝑡)𝑗, Re(𝜇𝑡)≥0.If𝜆1, 𝜆1,...,𝜆𝑟, 𝜆𝑟, ⇐⇒ 𝐴 𝑆=𝑆̃ 𝐽𝑗̃ 𝐽𝑗. =𝑆 𝜆𝑡 =𝑎𝑡 +𝑏𝑡𝑖, 𝑎𝑡,𝑏𝑡 ∈ R, 𝑏𝑡 ≥0, 𝑡=1,...,𝑟,arealleigenvaluesof 𝜎 complex representation 𝐴 ,then𝜇1,...,𝜇𝑟 are coneigenvalues 𝐴 ∼𝑐𝑠 𝐽𝑗 𝜆 =𝑎+𝑏𝑖 𝑎 ,𝑏 This means that .Let 𝑡 𝑡 𝑡 , 𝑡 𝑡 are real of quaternion matrix 𝐴,inwhich numbers, and 𝑏𝑡 ≥0,and𝐽=∑ ⊕𝐽𝑛 (𝜆𝑡)=∑ ⊕𝐽𝑛 (𝑎𝑡 −𝑏𝑡𝑖). 𝑡 𝑡 𝑡 𝑡 1 1 𝑐𝑠 𝐽𝑗 ∼ ∑ ⊕𝐽 (𝑏 +𝑎𝑗) 𝜇𝑡 =𝑏𝑡 +𝑎𝑡𝑗= (𝜆𝑡 − 𝜆𝑡)+ (𝜆𝑡 + 𝜆𝑡)𝑗, (19) Then by Proposition 2 we have 𝑡 𝑛𝑡 𝑡 𝑡 ,andwe 2𝑖 2 𝑐𝑠 𝑐𝑠 𝐴 ∼ ∑ ⊕𝐽 (𝑏 +𝑎𝑗) 𝐵 ∼ ∑ ⊕𝐽 (𝑏 +𝑎𝑗) −1 have 𝑡 𝑛𝑡 𝑡 𝑡 .Similarly 𝑡 𝑛𝑡 𝑡 𝑡 . 𝑝 𝑝̃ 𝜇 𝑝 𝑡=1,...,𝑟 𝑐𝑠 and for any nonzero quaternion , 𝑡 , ,areall Therefore 𝐴 ∼𝐵. coneigenvalues of quaternion matrix 𝐴. The statement above implies the following result. The following result comes immediately from Theorem 5 𝑛×𝑛 Theorem 5. Let 𝐴, 𝐵 ∈ H .Then𝐴 is consimilar to 𝐵 if and and the definition of complex representation in(2). 𝐴𝜎 only if complex representation matrix is similar to complex 𝑛×𝑛 𝜎 𝑐𝑠 𝜎 𝑠 𝜎 Corollary 10. 𝐴, 𝐵 ∈ C 𝐴 𝐵 representation matrix 𝐵 ;thatis,𝐴 ∼𝐵if and only if 𝐴 ∼𝐵 . Let .Then is consimilar to if [ 0−𝐴] [ 0−𝐵] and only if 𝐴0 is similar to 𝐵0 . Combining Proposition 3 and Theorem 5,wegetthefol- lowing result. 4. Algebraic Methods and Applications 𝑛×𝑛 Corollary 6. Let 𝐴, 𝐵 ∈ H .Then This section gives an algebraic method for relation between

𝑐𝑠 𝑠 consimilarity and similarity of quaternion matrices by means 𝐴 ∼𝐵⇐⇒𝐴𝑗∼𝐵𝑗 of complex representation and obtains an algebraic method (15) for coneigenvalues and coneigenvectors for quaternion matri- 𝑠 𝑠 𝜎 𝜎 ⇐⇒ 𝑗 𝐴 ∼𝑗𝐵⇐⇒𝑗𝐴∼𝐵𝑗⇐⇒𝐴 ∼𝐵 . ces as an application. 4 Journal of Applied Mathematics

𝑛×𝑛 𝑚×𝑚 𝜎 𝑠 𝜎 Let 𝐴∈H , 𝐵∈H . Suppose there exists a complex 𝐴 ∼𝐵. In which case, there exists a nonsingular complex 2𝑛×2𝑚 𝜎 𝜎 𝜎 𝜎 matrix 𝑇∈C such that 𝐴 𝑇=𝑇𝐵 .By(4) matrix 𝑇 such that 𝐴 𝑇=𝑇𝐵 ;let 𝐼 𝐴𝜎𝑇=𝑇𝐵𝜎 ⇐⇒ 𝐴 𝜎 (𝑄 𝑇𝑄𝑇 )=(𝑄𝑇𝑄𝑇 )𝐵𝜎. 1 𝑇 𝑛 𝑛 𝑚 𝑛 𝑚 (20) 𝑆= (𝐼𝑛,𝑗𝐼𝑛)(𝑇+𝑄𝑛𝑇𝑄𝑛 )[ ]𝑗. (28) 4 −𝑗𝐼𝑛 ̂ 𝑇 Let 𝑇 = (1/2)(𝑇𝑛 +𝑄 𝑇𝑄 ).Then 𝑚 Then 𝑆 is a quaternion matrix and 𝐴𝑆=𝑆𝐵̃ ,andif𝑆 is a 𝑐𝑠 𝜎 𝜎 𝐴 ∼𝐵 𝐴 𝑇̂ = 𝑇̂𝐵 . (21) nonsingular, then .

Let Corollary 12 gives an algebraic method for the relation between consimilarity and similarity of quaternion matrices 𝑇11 𝑇12 by means of complex representation and turns theory of 𝑇=[ ], (22) 𝑇21 𝑇22 consimilarityofquaternionmatricesintothatofordinary similarity of complex representation matrices. 𝑛×𝑚 where 𝑇𝑘𝑙 ∈ C . It is easy to get by direct calculation As a special case of Corollary 12, we give an algebraic method for relation between coneigenvalues and coneigen- 𝑇̂ −𝑇̂ vectors of a quaternion matrix and corresponding eigenval- ̂ 1 2 𝑇=[ ] , (23) ̂ ̂ ues and eigenvectors of the complex representation matrix as [𝑇2 𝑇1 ] follows.

𝑛×𝑛 where Corollary 13. Let 𝐴∈H , 𝜆∈H. Then there exists a vector 𝑛×1 1 1 𝛼∈H such that 𝐴𝛼=𝛼𝜆̃ if and only if there exists a vector 𝑇̂ = (𝑇 + 𝑇 ), 𝑇̂ = (𝑇 − 𝑇 ). 2𝑛×1 𝜎 𝜎 1 2 11 22 2 2 21 12 (24) 𝛽∈H such that 𝐴 𝛽=𝛽𝜆.Inwhichcase,thereexistsa 2𝑛×1 𝜎 𝜎 vector 𝛽∈H such that 𝐴 𝛽=𝛽𝜆 ;let From (23)weconstructacomplexmatrix𝑆 using the identi- 1 1 𝑗𝑇̂ = 𝑇̂ 𝑗 𝑗𝑇̂ 𝑗=−𝑇̂ 𝛼= (𝐼 ,𝑗𝐼 )(𝛽+𝑄 𝛽𝑄𝑇)[ ]𝑗. ties 𝑡 𝑡 and 𝑡 𝑡 4 𝑛 𝑛 𝑛 1 −𝑗 (29) 𝐼 ̂ ̂ 1 ̂ 𝑚 Then 𝐴𝛼=𝛼𝜆̃ . 𝑆=𝑇2 + 𝑇1𝑗= (𝐼𝑛,𝑗𝐼𝑛) 𝑇[ ] 2 −𝑗𝐼𝑚 (25) Now we give algebraic methods for finding coneigenval- 𝐼 1 𝑇 𝑚 ues and coneigenvectors of a quaternion matrix by means of = (𝐼𝑛,𝑗𝐼𝑛)(𝑇+𝑄𝑛𝑇𝑄𝑚)[ ]. complex representation. 4 −𝑗𝐼𝑚 𝑛×𝑛 𝜎 For 𝐴∈H ,if𝜇 is a complex eigenvalue of 𝐴 with 𝜎 2𝑛×1 𝜎 𝑆𝜎 = 𝑇̂ 𝑗𝐵𝑗 =−𝐵̃ 𝐴 𝛽=𝛽𝜇, 𝛽∈C , 𝛽 =0̸ ,thenby(11)and(12), 𝐴 𝛽=𝛽𝜇, Clearly . Therefore by (5)and ,(21)isequiv- 𝜎 𝑇 𝑇 alent to 𝐴 (𝑄𝑛 𝛽) = (𝑄𝑛 𝛽)𝜇,and 𝜎 𝜎 𝜎 𝜎 𝜎 𝜎 𝜇0 𝐴 𝑆 =𝑆 𝐵 ⇐⇒ ( 𝐴 𝑆)̃ =(𝑆𝐵)̃ 𝐴𝜎 (𝛽,𝑇 𝑄 𝛽) = (𝛽,𝑇 𝑄 𝛽) [ ]⇐⇒𝐴𝜎𝑇=𝑇(𝜇𝑗)𝜎, 𝑛 𝑛 0 𝜇 (26) ⇐⇒ 𝐴 𝑆=𝑆̃ 𝐵⇐⇒𝐴̃ 𝑆𝑗̃ = 𝑆𝑗𝐵. (30) 𝑇=(𝛽,𝑄𝑇𝛽) ∈ C2𝑛×2 The statement above implies the following result. in which 𝑛 .ByCorollary 13,let

𝑛×𝑛 𝑚×𝑚 1 𝑇 1 Proposition 11. Let 𝐴∈H , 𝐵∈H . Then there exists 𝛼= (𝐼𝑛,𝑗𝐼𝑛)(𝛽+𝑄𝑛𝛽𝑄1 )[ ]𝑗. (31) 𝑛×𝑚 4 −𝑗 a quaternion matrix 𝑆∈H such that 𝐴𝑆=𝑆𝐵̃ if and only 2𝑛×2𝑚 𝜎 if there exists a complex matrix 𝑇∈H such that 𝐴 𝑇= Then 𝐴𝛼=𝛼(̃ 𝜇𝑗).Thismeansthat𝜇𝑗 is a coneigenvalue 𝜎 𝐴 𝛼 =0̸ 𝛼 𝑇𝐵 . In which case, there exists a complex matrix 𝑇 such that of and if ,then is a coneigenvector related to 𝜎 𝜎 𝜇𝑗 𝐴 𝑇=𝑇𝐵 ;let coneigenvalue . By the statement above we get the following result. 1 𝐼𝑚 𝑆= (𝐼 ,𝑗𝐼 )(𝑇+𝑄 𝑇𝑄𝑇 )[ ]𝑗. 𝑛×𝑛 4 𝑛 𝑛 𝑛 𝑚 (27) Corollary 14. Let 𝐴∈H .If𝜇 is a complex eigenvalue of −𝑗𝐼𝑚 𝜎 complex representation 𝐴 and 𝛽 is eigenvector related to 𝜇 𝐴𝜎𝛽=𝛽𝜇 Then 𝑆 is a quaternion matrix and 𝐴𝑆=𝑆𝐵̃ . with ,let 1 1 𝛼= (𝐼 ,𝑗𝐼 )(𝛽+𝑄 𝛽𝑄𝑇)[ ]𝑗. By Theorem 5 and the statement above we have the 4 𝑛 𝑛 𝑛 1 −𝑗 (32) following result. 𝐴𝛼=𝛼(̃ 𝜇𝑗) 𝛼 =0̸ 𝜇𝑗 𝑛×𝑛 Then .Thismeansif ,then is a coneigen- Corollary 12. Let 𝐴, 𝐵 ∈ H .Then𝐴 is consimilar to 𝐵 if value of 𝐴 and 𝛼 is a coneigenvector related to coneigenvalue 𝜎 𝜎 𝑐𝑠 and only if 𝐴 is similar to 𝐵 ;thatis,𝐴 ∼𝐵if and only if 𝜇𝑗. Journal of Applied Mathematics 5

Moreover, let 𝜇=𝑎+𝑏𝑖, 𝑎, 𝑏 ∈ R,and𝑏≥0.Thenby [5]T.Jiang,X.Cheng,andL.Chen,“Analgebraicrelationbetween ̃ Proposition 2 and the fact that ((𝑎 + 𝑏𝑖)𝑗)(𝑖 + 𝑗) = (𝑖 + 𝑗)(−𝑏 + consimilarity and similarity of complex matrices and its appli- 𝑎𝑗) we have cations,” JournalofPhysicsA,vol.39,no.29,pp.9215–9222, 2006. 𝐴𝛽=𝛽(̃ 𝜇𝑗) ⇐⇒𝛽(𝑖+𝑗)) 𝐴( ̃ [6] S. L. Adler, “Quaternionic quantum field theory,” Physical (33) Review Letters,vol.55,no.8,pp.783–786,1985. =(𝛽(𝑖+𝑗))(𝑏+𝑎𝑗)⇐⇒𝐴𝛾=𝛾𝜆,̃ [7] S. L. Adler, “Quaternionic quantum field theory,” Communica- tions in Mathematical Physics,vol.104,no.4,pp.611–656,1986. in which 𝜆=𝑏+𝑎𝑗and 𝛾=𝛽(𝑖+𝑗).Thatis𝜆 is a coneigenvalue [8] S. L. Adler, “Scattering and decay theory for quaternionic quan- of quaternion matrix 𝐴,and𝛾 is a coneigenvector related to tum mechanics, and the structure of induced 𝑇 nonconserva- 𝜆. tion,” Physical Review D Particles and Fields,vol.37,no.12,pp. By Corollary 14 and the statement above we have the 3654–3662, 1988. following result. [9] S. L. Adler, Quaternionic Quantum Mechanics and Quantum Fields,OxfordUniversityPress,NewYork,NY,USA,1994. 𝑛×𝑛 Theorem 15. Let 𝐴∈H .If𝜇 is a complex eigenvalue of [10] A. J. Davies and B. H. J. McKellar, “Nonrelativistic quaternionic 𝜎 complex representation 𝐴 and 𝛽 is an eigenvector related to 𝜇 quantum mechanics in one dimension,” Physical Review A,vol. 𝜎 with 𝐴 𝛽=𝛽𝜇,let 40,no.8,pp.4209–4214,1989. [11] A. J. Davies and B. H. J. McKellar, “Observability of quaternionic 1 1 𝜆= (𝜇 − 𝜇) + (𝜇 + 𝜇) 𝑗, quantum mechanics,” Physical Review A,vol.46,no.7,pp.3671– 2𝑖 2 3675, 1992. 1 1 [12] A. G. Klein, “Schrodinger¨ inviolate: neutron optical searches for 𝛼= (𝐼 ,𝑗𝐼 )(𝛽+𝑄 𝛽𝑄𝑇)[ ]𝑗, 𝑛 𝑛 𝑛 1 −𝑗 violations of quantum mechanics,” Physica B,vol.151,no.1-2,pp. 4 44–49, 1988. 1 1 [13] A. Peres, “Proposed test for complex versus quaternion quan- 𝛾=𝛼(𝑖+𝑗)= (𝐼 ,𝑗𝐼 )(𝛽+𝑄 𝛽𝑄𝑇)[ ]𝑗(𝑖+𝑗). 4 𝑛 𝑛 𝑛 1 −𝑗 tum theory,” Physical Review Letters,vol.42,no.11,pp.683–686, 1979. (34) [14] P. Sutcliffe, “Instantons and the buckyball,” High Energy Then 𝐴𝛼=𝛼(̃ 𝜇𝑗) and 𝐴𝛾=𝛾𝜆̃ .Thesemeanthat(𝜇𝑗) and 𝜆 Phys.Theory,vol.16,p.157,2003. are two coneigenvalues of quaternion matrix 𝐴,andif𝛽 =0̸ , [15] S. de Leo and P. Rotelli, “Odd-dimensional translation between then 𝛼 and 𝛾 are corresponding coneigenvectors related to (𝜇𝑗) complex and quaternionic quantum mechanics,” Progress of −1 Theoretical Physics,vol.96,no.1,pp.247–255,1996. and 𝜆,respectively.Moreover𝑝̃ 𝜆𝑝 are right coneigenvalues of 𝐴,and𝛾𝑝̃ is corresponding coneigenvector for any nonzero [16] S. de Leo and G. Scolarici, “Right eigenvalue equation in quater- nionic quantum mechanics,” JournalofPhysicsA,vol.33,no.15, quaternion 𝑝. pp. 2971–2995, 2000. [17] S. de Leo, G. Scolarici, and L. Solombrino, “Quaternionic eigen- Conflict of Interests value problem,” Journal of Mathematical Physics,vol.43,no.11, pp. 5815–5829, 2002. The authors declare that there is no conflict of interests [18] H. Liping, “Consimilarity of quaternion matrices and complex regarding the publication of this paper. matrices,” Linear Algebra and Its Applications,vol.331,no.1–3, pp.21–30,2001. Acknowledgments [19] T. Jiang and S. Ling, “Algebraic methods for condiagonaliza- tion under consimilarity of quaternion matrices in quaternionic The authors would like to express their sincere thanks to the quantum mechanics,” Advances in Applied Clifford Algebras,vol. referees for the careful reading and very helpful comments on 23,no.2,pp.405–415,2013. the earlier versions of this paper. This paper is supported by [20] P. Lancaster and M. Tismenetsky, The Theory of Matrices with the National Natural Science Foundation of China (11301529 Applications, Academic Press, New York, NY, USA, 2nd edition, and 11301252) and Postdoctoral Science Foundation of China 1985. (2013M540472).

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