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, 276005, 2 College of Mathematics and System Science, of Science and Technology, , Shandong 266590, China 3 College of Mathematics and Statistics Science, Ludong University, , 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 ยฉ 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 ๐ด and ๐ต are both quaternion matrices of ๐‘›ร—๐‘›,theyaresaid โˆ’1 to be consimilar if ๐‘† ๐ด๐‘†=๐ตฬƒ holds for some ๐‘›ร—๐‘› nonsingular An antilinear operator ๐‘‡ is a mapping from one complex ๐‘  ๐‘๐‘  quaternion matrix ๐‘†.Write๐ด โˆผ๐ตif ๐ด is similar to ๐ต, ๐ด โˆผ๐ต vector space ๐‘‰ into another ๐‘Š, which is additive and conju- ๐‘๐‘  gate homogeneous, that is, for all ๐›ผ, ๐›ฝ โˆˆ๐‘‰ and any complex if ๐ด is consimilar to ๐ต,and๐ด โˆผ๐ตif ๐ด is permutation sim- number ๐‘Ž, ๐‘‡(๐›ผ + ๐›ฝ) = ๐‘‡(๐›ผ), +๐‘‡(๐›ฝ) ๐‘‡(๐‘Ž๐›ผ) = ๐‘Ž๐‘‡(๐›ผ) in which ilar to ๐ต. Permutation similarity is both a similarity and con- ๐‘Ž is the conjugate of ๐‘Ž.Two๐‘›ร—๐‘›complex matrices ๐ด, ๐ต are similarity relations. A quaternion ๐œ† is said to be the right โˆ’1 ฬƒ said to be consimilar if ๐‘† ๐ด๐‘†=๐ตfor some ๐‘›ร—๐‘›nonsingular coneigenvalueattributedto๐ด if the con-eigenequation ๐ด๐›ผ= complex matrix ๐‘†. Consimilarity of complex matrices arises ๐›ผ๐œ† 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 โŠ• R๐‘– the complex equivalent relation and has been studied [2โ€“5]. number field, and H = R โŠ• R๐‘–โŠ•R๐‘—โŠ•R๐‘˜ the quaternion 2 2 2 ๐‘šร—๐‘› In recent years, applications of quaternion matrices are field, where ๐‘–๐‘—=โˆ’๐‘—๐‘–=๐‘˜, ๐‘– =๐‘— =๐‘˜ =โˆ’1. F denotes ๐‘šร—๐‘› getting more and more important and extensive in quantum the set of all ๐‘šร—๐‘›matrices on a field F.For๐ดโˆˆH ,let ๐‘‡ mechanics, rigid mechanics, and control theory [6โ€“17]; it is ๐ด be the transpose of ๐ด and ๐ด the conjugate matrix of ๐ด. becoming more and more necessary to study the theory and For a quaternion ๐›ผ=๐‘Ž1 +๐‘Ž2๐‘–+๐‘Ž3๐‘—+๐‘Ž4๐‘˜ with ๐‘Ž๐‘ก โˆˆ R,we methods of quaternion matrices. In paper [18], the author use ๐›ผ=๐‘Ž1 โˆ’๐‘Ž2๐‘–โˆ’๐‘Ž3๐‘—โˆ’๐‘Ž4๐‘˜ and ๐›ผ=๐‘Žฬƒ 1 โˆ’๐‘Ž2๐‘–+๐‘Ž3๐‘—โˆ’๐‘Ž4๐‘˜ introduced concepts of consimilarity of quaternion matrices. todenotetheconjugateand๐‘—-conjugate of ๐›ผ,respectively. 2 Journal of Applied Mathematics

๐‘šร—๐‘› ๐‘›ร—๐‘› Let ๐ด=๐ด1 +๐ด2๐‘–+๐ด3๐‘—+๐ด4๐‘˜,where๐ด๐‘  โˆˆ R ; define Proposition 1. Let ๐ดโˆˆH .Then ๐ด=โˆ’๐‘—๐ด๐‘—=๐ดฬƒ โˆ’๐ด ๐‘–+๐ด ๐‘—โˆ’๐ด ๐‘˜ ๐‘— ๐ด 1 2 3 4 to be -conjugate of .It ๐ด๐œŽ ฬƒ ฬƒ (1) the real eigenvalues of complex representation is easy to verify that ๐‘—๐ด๐‘— =โˆ’๐ด, ๐ด=๐ด,and appear in pairs, and the imaginary eigenvalues of com- ๐œŽ ฬƒ ฬƒ ฬƒ ฬƒ ฬƒฬƒ plex representation ๐ด appear in conjugate pairs; that ๐ด+๐ต= ๐ด + ๐ต, ๐ด๐ถ = ๐ด๐ถ, (1) ๐œŽ is, if complex eigenvalue ๐œ† is an eigenvalue of ๐ด ,then ๐œŽ for any two ๐‘›ร—๐‘›matrices ๐ด, ๐ต and a ๐‘›ร—๐‘›matrix ๐ถ with complex eigenvalue ๐œ† is also an eigenvalue of ๐ด , quaternion entries. Obviously, ๐ด=ฬƒ ๐ด if ๐ด is a complex matrix. (2) the real Jordan blocks ๐ฝ๐‘Ÿ(๐œ†) of complex representation By means of real representation of complex matrices ๐œŽ ๐ด appear in pairs, and the imaginary Jordan blocks in paper [5], we studied the properties of consimilarity of ๐ฝ (๐œ†) ๐ด๐œŽ complex matrices and gave a relation between consimilarity ๐‘Ÿ of complex representation appear in conjugate pairs; that is, if ๐ฝ๐‘Ÿ(๐œ†) is a Jordan block of complex and similarity of complex matrices. This paper, by means of ๐ด๐œŽ ๐œ† a complex representation of a quaternion matrix, studies the representation related to complex eigenvalue ,then ๐ฝ (๐œ†) ๐ด๐œŽ relation between consimilarity and similarity of quaternion ๐‘Ÿ is a Jordan block of complex representation matrices and derives an algebraic relation between consim- related to complex eigenvalue ๐œ†. 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 ฬƒ ฬƒ of quaternion matrices. ๐‘—๐ด๐‘— =โˆ’๐ด, (๐ฝ๐‘Ÿ(๐‘Ž + ๐‘๐‘–))๐‘—(๐‘– + ๐‘—) = (๐‘–๐‘Ÿ +๐‘—)๐ฝ (โˆ’๐‘ + ๐‘Ž๐‘—) and ((๐ด1 + ฬƒ ๐ด2๐‘–)๐‘—)(๐‘– + ๐‘—) = (๐‘– + ๐‘—)(โˆ’๐ด2 +๐ด1๐‘—) by direct calculation for 2. Preliminaries real numbers ๐‘Ž and ๐‘,andrealmatrices๐ด1 and ๐ด2. ๐ด ๐‘›ร—๐‘š For a quaternion matrix of dimension , denote by Proposition 2. Let ๐ด be a quaternion matrix. Then ๐ด=๐ต1 +๐ต2๐‘–+๐ต3๐‘—+๐ต4๐‘˜โ‰ก๐ด1 +๐ด2๐‘—,where๐ด1 =๐ต1 +๐ต2๐‘–, ฬƒ ฬƒ ๐ด2 =๐ต3 +๐ต4๐‘–. The complex representation of the quaternion (1) ๐ด๐‘— =๐‘—๐ด;thismeansthat๐ด is consimilar to ๐ด;thatis, ๐ด ๐‘๐‘  matrix is defined [19]tobe ๐ด โˆผ ๐ดฬƒ; ๐ด โˆ’๐ด ๐‘๐‘  ๐œŽ 2 1 2๐‘›ร—2๐‘š ๐ฝ=๐ฝ(๐‘Ž + ๐‘๐‘–) ๐‘Ž, ๐‘ โˆˆ R ๐ฝ๐‘— โˆผ ๐ด =[ ]โˆˆC ; (2) (2) if ๐‘Ÿ is a Jordan block, ,then ๐ด1 ๐ด2 ๐ฝ๐‘Ÿ(โˆ’๐‘ + ๐‘Ž๐‘—); ๐œŽ ๐‘›ร—๐‘› the complex matrix ๐ด is known as complex representation of (3) if ๐ด=๐ด1 +๐ด2๐‘– is a complex matrix, ๐ด1,๐ด2 โˆˆ R , ๐œŽ the quaternion matrix ๐ด.Itiseasytoverifythat๐ด๐œŽ =(๐ด)ฬƒ . then ๐‘›ร—๐‘š ๐‘šร—๐‘ Let ๐ด, ๐ต โˆˆ H , ๐ถโˆˆH , ๐‘ŽโˆˆR. Then by the definition ๐‘๐‘  ๐‘๐‘  ๐ด๐‘— โˆผโˆ’๐ด2 +๐ด1๐‘—, ๐ด๐‘— โˆผ(๐ด1 +๐ด2๐‘—) ๐‘—. (8) ofcomplexrepresentationweeasilygetthefollowingresults: ๐œŽ ๐œŽ ๐œŽ ๐œŽ ๐œŽ (๐ด+๐ต) =๐ด +๐ต , (๐‘Ž๐ด) =๐‘Ž๐ด ; (3) Based on the quaternion field H,theauthorin[18]gave the following results. ๐‘‡ ๐œŽ ๐œŽ ๐‘„ ๐ด ๐‘„ = ๐ด๐œŽ =(๐ด)ฬƒ ; (4) ๐‘› ๐‘š ๐‘›ร—๐‘› Proposition 3 (see [18]). If ๐ด, ๐ต โˆˆ H ,then ๐œŽ ๐œŽ ๐œŽ ๐œŽ ฬƒ ๐œŽ ฬƒ ๐œŽ ๐œŽ (๐ด๐ถ) =๐ด ๐‘„๐‘š๐ถ =๐ด (๐ถ) ๐‘„๐‘ =๐‘„๐‘›(๐ด) ๐ถ , (5) ๐‘๐‘  ๐‘  ๐‘  ๐‘  ๐ด โˆผ ๐ตโ‡โ‡’๐‘—๐ดโˆผ ๐‘—๐ต โ‡โ‡’ ๐ด๐‘— โˆผ ๐ต๐‘— โ‡โ‡’ ๐‘—๐ด โˆผ ๐ต๐‘—. (9) 0๐ผ๐‘ก in which ๐‘„๐‘ก =[ ] with ๐ผ๐‘ก being the identity matrix, โˆ’๐ผ๐‘ก 0 ๐‘›ร—๐‘› 2 ๐‘‡ ๐‘‡ Proposition 4 (see [18]). If ๐ดโˆˆH ,then and ๐‘„๐‘ก =โˆ’๐ผ2๐‘ก, ๐‘„๐‘ก๐‘„๐‘ก =๐‘„๐‘ก ๐‘„๐‘ก =๐ผ2๐‘ก.Clearly,from(5)we know that a quaternion matrix ๐ด is nonsingular if and only if ๐‘๐‘  ๐‘ ๐œŽ ๐ด โˆผ๐ฝ๐‘› (๐œ†1)โŠ•๐ฝ๐‘› (๐œ†2)โŠ•โ‹…โ‹…โ‹…โŠ•๐ฝ๐‘› (๐œ†๐‘Ÿ)โ‰ก๐ฝ , (10) complex representation matrix ๐ด is nonsingular. 1 2 ๐‘Ÿ ๐ด ๐‘›ร—๐‘› 2๐‘›ร—1 ๐œŽ For ๐ดโˆˆH and ๐›ผโˆˆC ,if๐ด ๐›ผ=๐œ†๐›ผ,thenby(4)we in which ๐œ†๐‘ก =๐‘Ž๐‘ก +๐‘๐‘ก๐‘— are right coneigenvalues of ๐ด, ๐‘Ž๐‘ก,๐‘๐‘ก are ๐‘ have real numbers, and ๐‘Ž๐‘ก โ‰ฅ0. ๐ฝ๐ด is uniquely determined by ๐ด up ๐œŽ ๐‘‡ ๐‘‡ ๐œŽ ๐‘‡ ๐‘‡ ๐‘ to the order of Jordan blocks ๐ฝ๐‘› (๐œ†๐‘ก),and๐ฝ๐ด is said to be the ๐ด (๐‘„๐‘› ๐›ผ) =๐œ†(๐‘„๐‘› ๐›ผ) โ‡โ‡’ ๐ด (๐‘„๐‘› ๐›ผ) = ๐œ†(๐‘„๐‘› ๐›ผ) . (6) ๐‘ก ๐ด ๐œŽ Jordan canonical form of under consimilarity. This means that the eigenvalues of complex representation ๐ด appear in conjugate pairs. 3. An Algebraic Relation between In the same manner, for a Jordan block ๐ฝ๐‘Ÿ(๐œ†) of an ๐œŽ eigenvalue ๐œ†,if๐ด ๐‘‹=๐‘‹๐ฝ๐‘Ÿ(๐œ†),thenby(4)wehave Consimilarity and Similarity ๐œŽ ๐‘‡ ๐‘‡ ๐œŽ ๐‘‡ ๐‘‡ ๐ด (๐‘„๐‘› ๐‘‹) = (๐‘„๐‘› ๐‘‹)๐‘Ÿ ๐ฝ (๐œ†) โ‡โ‡’ ๐ด (๐‘„๐‘› ๐‘‹) = (๐‘„๐‘› ๐‘‹)๐‘Ÿ ๐ฝ (๐œ†) . This section gives an algebraic relation between consimilarity and similarity of quaternion matrices by means of the com- (7) plex representation in (2). ๐ด, ๐ต โˆˆ H๐‘›ร—๐‘› ๐ด And this means Jordan blocks ๐ฝ๐‘Ÿ(๐œ†) of complex representa- Let be two quaternion matrices. If is ๐œŽ tion ๐ด appear in conjugate pairs. consimilar to ๐ต, then there exists a nonsingular quaternion ฬƒ ๐œŽ ๐œŽ ๐œŽ ๐œŽ From the statement above we have the following result. matrix ๐‘† such that ๐ด๐‘† = ๐‘†๐ต,by(5) ๐ด ๐‘„๐‘›๐‘† =๐‘„๐‘›๐‘† ๐ต . Journal of Applied Mathematics 3

๐œŽ This means that if ๐ด is consimilar to ๐ต,then๐ด is similar to The proof of the Theorem 5 is constructive, and the ๐œŽ ๐ต . following results come from the proof above. ๐ด๐œŽ ๐ต๐œŽ ๐ด๐œŽ ๐ต๐œŽ Conversely, if is similar to ,then and have the Corollary 7. ๐ดโˆˆH๐‘›ร—๐‘› ๐œ† , ๐œ† ,...,๐œ† , ๐œ† Let . Then the following statements are same eigenvalues. By Proposition 1,let 1 1 ๐‘Ÿ ๐‘Ÿ be equivalent. all eigenvalues and ๐ฝ๐‘› (๐œ†1), ๐ฝ๐‘› (๐œ†1),...,๐ฝ๐‘› (๐œ†๐‘Ÿ), ๐ฝ๐‘› (๐œ†๐‘Ÿ) corre- 1 1 ๐‘Ÿ ๐‘Ÿ ๐ด๐œŽ sponding Jordan blocks. There exists a complex and full-rank (1) The Jordan canonical form of complex matrix under ๐‘‹ ๐ด๐œŽ๐‘‹ =๐‘‹๐ฝ (๐œ† ) similarity is matrix ๐‘— by [20,chapter6.7]suchthat ๐‘ก ๐‘ก ๐‘›๐‘ก ๐‘ก , ๐‘ก=1,...,๐‘Ÿ.Thenby(4), we have ๐ฝ=โˆ‘ โŠ•๐ฝ๐‘› (๐œ†๐‘ก)โŠ•โˆ‘ โŠ•๐ฝ๐‘› (๐œ†๐‘ก), ๐‘ก ๐‘ก (16) ๐‘ก ๐‘ก ๐ด๐œŽ๐‘‹ =๐‘‹๐ฝ (๐œ† ), ๐ด๐œŽ (๐‘„๐‘‡๐‘‹ )=(๐‘„๐‘‡๐‘‹ )๐ฝ (๐œ† ), ๐‘ก ๐‘ก ๐‘›๐‘ก ๐‘ก ๐‘› ๐‘ก ๐‘› ๐‘ก ๐‘›๐‘ก ๐‘ก where ๐œ†๐‘ก =๐‘Ž๐‘ก +๐‘๐‘ก๐‘–, ๐‘Ž๐‘ก,๐‘๐‘ก โˆˆ R, ๐‘๐‘ก โ‰ฅ0, ๐‘ก = 1,...,๐‘Ÿ,are (11) ๐œŽ eigenvalues of ๐ด . and (11)isequivalentto (2) The Jordan canonical form of quaternion matrix ๐ด ๐ฝ (๐œ† )0 under consimilarity is ๐œŽ ๐‘‡ ๐‘‡ ๐‘›๐‘ก ๐‘ก ๐ด (๐‘‹๐‘ก,๐‘„๐‘› ๐‘‹๐‘ก)=(๐‘‹๐‘ก,๐‘„๐‘› ๐‘‹๐‘ก)[ ]. (12) ๐‘ 0๐ฝ๐‘› (๐œ†๐‘ก) ๐ฝ = โˆ‘ โŠ•๐ฝ๐‘› (๐œ‡๐‘ก), ๐‘ก ๐‘ก (17) ๐‘ก Therefore there exists a nonsingular complex matrix ๐‘‡= where ๐œ‡๐‘ก =๐‘๐‘ก +๐‘Ž๐‘ก๐‘—, ๐‘Ž๐‘ก,๐‘๐‘ก โˆˆ R, ๐‘๐‘ก โ‰ฅ0, ๐‘ก = 1,...,๐‘Ÿ,are (๐‘,๐‘‡ ๐‘„ ๐‘) โˆˆ C2๐‘›ร—2๐‘› ๐‘โˆˆC2๐‘›ร—๐‘› ๐‘› , such that coneigenvalues of ๐ด.

๐‘›ร—๐‘› โˆ‘ โŠ•๐ฝ๐‘› (๐œ†๐‘ก)0 Corollary 8. ๐ดโˆˆH ๐œŽ [ ๐‘ก ] Let .Then ๐‘‡โˆ’1๐ด๐œŽ๐‘‡ = (๐ฝ๐‘—) = [ ๐‘ก ] , (13) ๐‘๐‘  0 โˆ‘ โŠ•๐ฝ๐‘› (๐œ†๐‘ก) ๐‘ ๐‘ก ๐ด โˆผ๐ฝ (๐œ‡ )โŠ•๐ฝ (๐œ‡ )โ‹…โ‹…โ‹…โŠ•๐ฝ (๐œ‡ )โ‰ก๐ฝ, (18) [ ๐‘ก ] ๐‘›1 1 ๐‘›2 2 ๐‘›๐‘Ÿ ๐‘Ÿ ๐œ‡ =๐‘+๐‘Ž๐‘— = (1/2๐‘–)(๐œ† โˆ’ ๐œ† ) + (1/2)(๐œ† + ๐œ† )๐‘— in which ๐ฝ=โˆ‘๐‘ก โŠ•๐ฝ๐‘› (๐œ†๐‘ก), ๐œ†๐‘ก =๐‘Ž๐‘ก +๐‘๐‘ก๐‘–, ๐‘Ž๐‘ก,๐‘๐‘ก are real numbers, where ๐‘ก ๐‘ก ๐‘ก ๐‘ก ๐‘ก ๐‘ก ๐‘ก are ๐‘ก ๐ด ๐œ† =๐‘Ž+๐‘ ๐‘– ๐‘Ž ,๐‘ โˆˆ R ๐‘ โ‰ฅ0 ๐‘ก=1,...,๐‘Ÿ and ๐‘๐‘ก โ‰ฅ0. coneigenvalues of , ๐‘ก ๐‘ก ๐‘ก , ๐‘ก ๐‘ก , ๐‘ก , , ๐‘ 1 ๐‘›ร—๐‘› ๐œ† , ๐œ† ,...,๐œ† , ๐œ† Let ๐‘=[๐‘ ], ๐‘1,๐‘2 โˆˆ C ,and๐‘†=๐‘2 +๐‘1๐‘—.Then in which 1 1 ๐‘Ÿ ๐‘Ÿ are all eigenvalues of complex rep- 2 ๐œŽ ๐‘1 โˆ’๐‘2 ๐œŽ ๐ด ๐ฝ (๐œ† ), ๐ฝ (๐œ† ),...,๐ฝ (๐œ† ), ๐ฝ (๐œ† ) ๐‘‡=[ ]=๐‘† ๐‘‡ resentation ,and ๐‘›1 1 ๐‘›1 1 ๐‘›๐‘Ÿ ๐‘Ÿ ๐‘›๐‘Ÿ ๐‘Ÿ corre- ๐‘ ๐‘ . From the nonsingular complex matrix , ๐‘ 2 1 sponding Jordan blocks in (16). ๐ฝ is uniquely determined by ๐ด we get ๐‘† is a nonsingular quaternion matrix; by (1)and(5), ๐ฝ (๐œ‡ ) ๐ฝ๐‘ up to the order of Jordan blocks ๐‘›๐‘ก ๐‘ก ,and is said to be the (13)isequivalentto Jordan canonical form of ๐ด under consimilarity. ๐œŽ ๐œŽ ๐œŽ ๐œŽ ๐œŽ ๐œŽ ฬƒ ฬƒ ๐‘›ร—๐‘› ๐ด ๐‘† =๐‘† (๐ฝ๐‘—) โ‡โ‡’ ( ๐ด ๐‘†) =(๐‘†(๐ฝ๐‘—)) Corollary 9. Let ๐ดโˆˆH .Then๐ด has at least a coneigen- (14) value ๐œ‡๐‘ก = Re(๐œ‡๐‘ก)+Im(๐œ‡๐‘ก)๐‘—, Re(๐œ‡๐‘ก)โ‰ฅ0.If๐œ†1, ๐œ†1,...,๐œ†๐‘Ÿ, ๐œ†๐‘Ÿ, โ‡โ‡’ ๐ด ๐‘†=๐‘†ฬƒ ๐ฝ๐‘—ฬƒ ๐ฝ๐‘—. =๐‘† ๐œ†๐‘ก =๐‘Ž๐‘ก +๐‘๐‘ก๐‘–, ๐‘Ž๐‘ก,๐‘๐‘ก โˆˆ R, ๐‘๐‘ก โ‰ฅ0, ๐‘ก=1,...,๐‘Ÿ,arealleigenvaluesof ๐œŽ complex representation ๐ด ,then๐œ‡1,...,๐œ‡๐‘Ÿ are coneigenvalues ๐ด โˆผ๐‘๐‘  ๐ฝ๐‘— ๐œ† =๐‘Ž+๐‘๐‘– ๐‘Ž ,๐‘ This means that .Let ๐‘ก ๐‘ก ๐‘ก , ๐‘ก ๐‘ก are real of quaternion matrix ๐ด,inwhich numbers, and ๐‘๐‘ก โ‰ฅ0,and๐ฝ=โˆ‘ โŠ•๐ฝ๐‘› (๐œ†๐‘ก)=โˆ‘ โŠ•๐ฝ๐‘› (๐‘Ž๐‘ก โˆ’๐‘๐‘ก๐‘–). ๐‘ก ๐‘ก ๐‘ก ๐‘ก 1 1 ๐‘๐‘  ๐ฝ๐‘— โˆผ โˆ‘ โŠ•๐ฝ (๐‘ +๐‘Ž๐‘—) ๐œ‡๐‘ก =๐‘๐‘ก +๐‘Ž๐‘ก๐‘—= (๐œ†๐‘ก โˆ’ ๐œ†๐‘ก)+ (๐œ†๐‘ก + ๐œ†๐‘ก)๐‘—, (19) Then by Proposition 2 we have ๐‘ก ๐‘›๐‘ก ๐‘ก ๐‘ก ,andwe 2๐‘– 2 ๐‘๐‘  ๐‘๐‘  ๐ด โˆผ โˆ‘ โŠ•๐ฝ (๐‘ +๐‘Ž๐‘—) ๐ต โˆผ โˆ‘ โŠ•๐ฝ (๐‘ +๐‘Ž๐‘—) โˆ’1 have ๐‘ก ๐‘›๐‘ก ๐‘ก ๐‘ก .Similarly ๐‘ก ๐‘›๐‘ก ๐‘ก ๐‘ก . ๐‘ ๐‘ฬƒ ๐œ‡ ๐‘ ๐‘ก=1,...,๐‘Ÿ ๐‘๐‘  and for any nonzero quaternion , ๐‘ก , ,areall Therefore ๐ด โˆผ๐ต. coneigenvalues of quaternion matrix ๐ด. The statement above implies the following result. The following result comes immediately from Theorem 5 ๐‘›ร—๐‘› Theorem 5. Let ๐ด, ๐ต โˆˆ H .Then๐ด is consimilar to ๐ต if and and the definition of complex representation in(2). ๐ด๐œŽ only if complex representation matrix is similar to complex ๐‘›ร—๐‘› ๐œŽ ๐‘๐‘  ๐œŽ ๐‘  ๐œŽ Corollary 10. ๐ด, ๐ต โˆˆ C ๐ด ๐ต representation matrix ๐ต ;thatis,๐ด โˆผ๐ตif and only if ๐ด โˆผ๐ต . Let .Then is consimilar to if [ 0โˆ’๐ด] [ 0โˆ’๐ต] and only if ๐ด0 is similar to ๐ต0 . Combining Proposition 3 and Theorem 5,wegetthefol- lowing result. 4. Algebraic Methods and Applications ๐‘›ร—๐‘› Corollary 6. Let ๐ด, ๐ต โˆˆ H .Then This section gives an algebraic method for relation between

๐‘๐‘  ๐‘  consimilarity and similarity of quaternion matrices by means ๐ด โˆผ๐ตโ‡โ‡’๐ด๐‘—โˆผ๐ต๐‘— of complex representation and obtains an algebraic method (15) for coneigenvalues and coneigenvectors for quaternion matri- ๐‘  ๐‘  ๐œŽ ๐œŽ โ‡โ‡’ ๐‘— ๐ด โˆผ๐‘—๐ตโ‡โ‡’๐‘—๐ดโˆผ๐ต๐‘—โ‡โ‡’๐ด โˆผ๐ต . ces as an application. 4 Journal of Applied Mathematics

๐‘›ร—๐‘› ๐‘šร—๐‘š ๐œŽ ๐‘  ๐œŽ Let ๐ดโˆˆH , ๐ตโˆˆH . Suppose there exists a complex ๐ด โˆผ๐ต. In which case, there exists a nonsingular complex 2๐‘›ร—2๐‘š ๐œŽ ๐œŽ ๐œŽ ๐œŽ matrix ๐‘‡โˆˆC such that ๐ด ๐‘‡=๐‘‡๐ต .By(4) matrix ๐‘‡ such that ๐ด ๐‘‡=๐‘‡๐ต ;let ๐ผ ๐ด๐œŽ๐‘‡=๐‘‡๐ต๐œŽ โ‡โ‡’ ๐ด ๐œŽ (๐‘„ ๐‘‡๐‘„๐‘‡ )=(๐‘„๐‘‡๐‘„๐‘‡ )๐ต๐œŽ. 1 ๐‘‡ ๐‘› ๐‘› ๐‘š ๐‘› ๐‘š (20) ๐‘†= (๐ผ๐‘›,๐‘—๐ผ๐‘›)(๐‘‡+๐‘„๐‘›๐‘‡๐‘„๐‘› )[ ]๐‘—. (28) 4 โˆ’๐‘—๐ผ๐‘› ฬ‚ ๐‘‡ Let ๐‘‡ = (1/2)(๐‘‡๐‘› +๐‘„ ๐‘‡๐‘„ ).Then ๐‘š Then ๐‘† is a quaternion matrix and ๐ด๐‘†=๐‘†๐ตฬƒ ,andif๐‘† is a ๐‘๐‘  ๐œŽ ๐œŽ ๐ด โˆผ๐ต ๐ด ๐‘‡ฬ‚ = ๐‘‡ฬ‚๐ต . (21) nonsingular, then .

Let Corollary 12 gives an algebraic method for the relation between consimilarity and similarity of quaternion matrices ๐‘‡11 ๐‘‡12 by means of complex representation and turns theory of ๐‘‡=[ ], (22) ๐‘‡21 ๐‘‡22 consimilarityofquaternionmatricesintothatofordinary similarity of complex representation matrices. ๐‘›ร—๐‘š where ๐‘‡๐‘˜๐‘™ โˆˆ 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- ๐‘‡ฬ‚ โˆ’๐‘‡ฬ‚ vectors of a quaternion matrix and corresponding eigenval- ฬ‚ 1 2 ๐‘‡=[ ] , (23) ฬ‚ ฬ‚ ues and eigenvectors of the complex representation matrix as [๐‘‡2 ๐‘‡1 ] follows.

๐‘›ร—๐‘› where Corollary 13. Let ๐ดโˆˆH , ๐œ†โˆˆH. Then there exists a vector ๐‘›ร—1 1 1 ๐›ผโˆˆH such that ๐ด๐›ผ=๐›ผ๐œ†ฬƒ if and only if there exists a vector ๐‘‡ฬ‚ = (๐‘‡ + ๐‘‡ ), ๐‘‡ฬ‚ = (๐‘‡ โˆ’ ๐‘‡ ). 2๐‘›ร—1 ๐œŽ ๐œŽ 1 2 11 22 2 2 21 12 (24) ๐›ฝโˆˆH such that ๐ด ๐›ฝ=๐›ฝ๐œ†.Inwhichcase,thereexistsa 2๐‘›ร—1 ๐œŽ ๐œŽ vector ๐›ฝโˆˆH such that ๐ด ๐›ฝ=๐›ฝ๐œ† ;let From (23)weconstructacomplexmatrix๐‘† using the identi- 1 1 ๐‘—๐‘‡ฬ‚ = ๐‘‡ฬ‚ ๐‘— ๐‘—๐‘‡ฬ‚ ๐‘—=โˆ’๐‘‡ฬ‚ ๐›ผ= (๐ผ ,๐‘—๐ผ )(๐›ฝ+๐‘„ ๐›ฝ๐‘„๐‘‡)[ ]๐‘—. ties ๐‘ก ๐‘ก and ๐‘ก ๐‘ก 4 ๐‘› ๐‘› ๐‘› 1 โˆ’๐‘— (29) ๐ผ ฬ‚ ฬ‚ 1 ฬ‚ ๐‘š Then ๐ด๐›ผ=๐›ผ๐œ†ฬƒ . ๐‘†=๐‘‡2 + ๐‘‡1๐‘—= (๐ผ๐‘›,๐‘—๐ผ๐‘›) ๐‘‡[ ] 2 โˆ’๐‘—๐ผ๐‘š (25) Now we give algebraic methods for finding coneigenval- ๐ผ 1 ๐‘‡ ๐‘š ues and coneigenvectors of a quaternion matrix by means of = (๐ผ๐‘›,๐‘—๐ผ๐‘›)(๐‘‡+๐‘„๐‘›๐‘‡๐‘„๐‘š)[ ]. complex representation. 4 โˆ’๐‘—๐ผ๐‘š ๐‘›ร—๐‘› ๐œŽ For ๐ดโˆˆH ,if๐œ‡ is a complex eigenvalue of ๐ด with ๐œŽ 2๐‘›ร—1 ๐œŽ ๐‘†๐œŽ = ๐‘‡ฬ‚ ๐‘—๐ต๐‘— =โˆ’๐ตฬƒ ๐ด ๐›ฝ=๐›ฝ๐œ‡, ๐›ฝโˆˆC , ๐›ฝ =0ฬธ ,thenby(11)and(12), ๐ด ๐›ฝ=๐›ฝ๐œ‡, Clearly . Therefore by (5)and ,(21)isequiv- ๐œŽ ๐‘‡ ๐‘‡ alent to ๐ด (๐‘„๐‘› ๐›ฝ) = (๐‘„๐‘› ๐›ฝ)๐œ‡,and ๐œŽ ๐œŽ ๐œŽ ๐œŽ ๐œŽ ๐œŽ ๐œ‡0 ๐ด ๐‘† =๐‘† ๐ต โ‡โ‡’ ( ๐ด ๐‘†)ฬƒ =(๐‘†๐ต)ฬƒ ๐ด๐œŽ (๐›ฝ,๐‘‡ ๐‘„ ๐›ฝ) = (๐›ฝ,๐‘‡ ๐‘„ ๐›ฝ) [ ]โ‡โ‡’๐ด๐œŽ๐‘‡=๐‘‡(๐œ‡๐‘—)๐œŽ, ๐‘› ๐‘› 0 ๐œ‡ (26) โ‡โ‡’ ๐ด ๐‘†=๐‘†ฬƒ ๐ตโ‡โ‡’๐ดฬƒ ๐‘†๐‘—ฬƒ = ๐‘†๐‘—๐ต. (30) ๐‘‡=(๐›ฝ,๐‘„๐‘‡๐›ฝ) โˆˆ C2๐‘›ร—2 The statement above implies the following result. in which ๐‘› .ByCorollary 13,let

๐‘›ร—๐‘› ๐‘šร—๐‘š 1 ๐‘‡ 1 Proposition 11. Let ๐ดโˆˆH , ๐ตโˆˆH . Then there exists ๐›ผ= (๐ผ๐‘›,๐‘—๐ผ๐‘›)(๐›ฝ+๐‘„๐‘›๐›ฝ๐‘„1 )[ ]๐‘—. (31) ๐‘›ร—๐‘š 4 โˆ’๐‘— a quaternion matrix ๐‘†โˆˆH such that ๐ด๐‘†=๐‘†๐ตฬƒ if and only 2๐‘›ร—2๐‘š ๐œŽ if there exists a complex matrix ๐‘‡โˆˆH such that ๐ด ๐‘‡= Then ๐ด๐›ผ=๐›ผ(ฬƒ ๐œ‡๐‘—).Thismeansthat๐œ‡๐‘— is a coneigenvalue ๐œŽ ๐ด ๐›ผ =0ฬธ ๐›ผ ๐‘‡๐ต . In which case, there exists a complex matrix ๐‘‡ such that of and if ,then is a coneigenvector related to ๐œŽ ๐œŽ ๐œ‡๐‘— ๐ด ๐‘‡=๐‘‡๐ต ;let coneigenvalue . By the statement above we get the following result. 1 ๐ผ๐‘š ๐‘†= (๐ผ ,๐‘—๐ผ )(๐‘‡+๐‘„ ๐‘‡๐‘„๐‘‡ )[ ]๐‘—. ๐‘›ร—๐‘› 4 ๐‘› ๐‘› ๐‘› ๐‘š (27) Corollary 14. Let ๐ดโˆˆH .If๐œ‡ is a complex eigenvalue of โˆ’๐‘—๐ผ๐‘š ๐œŽ complex representation ๐ด and ๐›ฝ is eigenvector related to ๐œ‡ ๐ด๐œŽ๐›ฝ=๐›ฝ๐œ‡ Then ๐‘† is a quaternion matrix and ๐ด๐‘†=๐‘†๐ตฬƒ . with ,let 1 1 ๐›ผ= (๐ผ ,๐‘—๐ผ )(๐›ฝ+๐‘„ ๐›ฝ๐‘„๐‘‡)[ ]๐‘—. By Theorem 5 and the statement above we have the 4 ๐‘› ๐‘› ๐‘› 1 โˆ’๐‘— (32) following result. ๐ด๐›ผ=๐›ผ(ฬƒ ๐œ‡๐‘—) ๐›ผ =0ฬธ ๐œ‡๐‘— ๐‘›ร—๐‘› Then .Thismeansif ,then is a coneigen- Corollary 12. Let ๐ด, ๐ต โˆˆ H .Then๐ด is consimilar to ๐ต if value of ๐ด and ๐›ผ is a coneigenvector related to coneigenvalue ๐œŽ ๐œŽ ๐‘๐‘  and only if ๐ด is similar to ๐ต ;thatis,๐ด โˆผ๐ตif and only if ๐œ‡๐‘—. Journal of Applied Mathematics 5

Moreover, let ๐œ‡=๐‘Ž+๐‘๐‘–, ๐‘Ž, ๐‘ โˆˆ R,and๐‘โ‰ฅ0.Thenby [5]T.Jiang,X.Cheng,andL.Chen,โ€œAnalgebraicrelationbetween ฬƒ Proposition 2 and the fact that ((๐‘Ž + ๐‘๐‘–)๐‘—)(๐‘– + ๐‘—) = (๐‘– + ๐‘—)(โˆ’๐‘ + consimilarity and similarity of complex matrices and its appli- ๐‘Ž๐‘—) we have cations,โ€ JournalofPhysicsA,vol.39,no.29,pp.9215โ€“9222, 2006. ๐ด๐›ฝ=๐›ฝ(ฬƒ ๐œ‡๐‘—) โ‡โ‡’๐›ฝ(๐‘–+๐‘—)) ๐ด( ฬƒ [6] S. L. Adler, โ€œQuaternionic quantum field theory,โ€ Physical (33) Review Letters,vol.55,no.8,pp.783โ€“786,1985. =(๐›ฝ(๐‘–+๐‘—))(๐‘+๐‘Ž๐‘—)โ‡โ‡’๐ด๐›พ=๐›พ๐œ†,ฬƒ [7] S. L. Adler, โ€œQuaternionic quantum field theory,โ€ Communica- tions in Mathematical Physics,vol.104,no.4,pp.611โ€“656,1986. in which ๐œ†=๐‘+๐‘Ž๐‘—and ๐›พ=๐›ฝ(๐‘–+๐‘—).Thatis๐œ† is a coneigenvalue [8] S. L. Adler, โ€œScattering and decay theory for quaternionic quan- of quaternion matrix ๐ด,and๐›พ is a coneigenvector related to tum mechanics, and the structure of induced ๐‘‡ nonconserva- ๐œ†. tion,โ€ Physical Review D Particles and Fields,vol.37,no.12,pp. By Corollary 14 and the statement above we have the 3654โ€“3662, 1988. following result. [9] S. L. Adler, Quaternionic Quantum Mechanics and Quantum Fields,OxfordUniversityPress,NewYork,NY,USA,1994. ๐‘›ร—๐‘› Theorem 15. Let ๐ดโˆˆH .If๐œ‡ is a complex eigenvalue of [10] A. J. Davies and B. H. J. McKellar, โ€œNonrelativistic quaternionic ๐œŽ complex representation ๐ด and ๐›ฝ is an eigenvector related to ๐œ‡ quantum mechanics in one dimension,โ€ Physical Review A,vol. ๐œŽ with ๐ด ๐›ฝ=๐›ฝ๐œ‡,let 40,no.8,pp.4209โ€“4214,1989. [11] A. J. Davies and B. H. J. McKellar, โ€œObservability of quaternionic 1 1 ๐œ†= (๐œ‡ โˆ’ ๐œ‡) + (๐œ‡ + ๐œ‡) ๐‘—, quantum mechanics,โ€ Physical Review A,vol.46,no.7,pp.3671โ€“ 2๐‘– 2 3675, 1992. 1 1 [12] A. G. Klein, โ€œSchrodingerยจ inviolate: neutron optical searches for ๐›ผ= (๐ผ ,๐‘—๐ผ )(๐›ฝ+๐‘„ ๐›ฝ๐‘„๐‘‡)[ ]๐‘—, ๐‘› ๐‘› ๐‘› 1 โˆ’๐‘— violations of quantum mechanics,โ€ Physica B,vol.151,no.1-2,pp. 4 44โ€“49, 1988. 1 1 [13] A. Peres, โ€œProposed test for complex versus quaternion quan- ๐›พ=๐›ผ(๐‘–+๐‘—)= (๐ผ ,๐‘—๐ผ )(๐›ฝ+๐‘„ ๐›ฝ๐‘„๐‘‡)[ ]๐‘—(๐‘–+๐‘—). 4 ๐‘› ๐‘› ๐‘› 1 โˆ’๐‘— tum theory,โ€ Physical Review Letters,vol.42,no.11,pp.683โ€“686, 1979. (34) [14] P. Sutcliffe, โ€œInstantons and the buckyball,โ€ High Energy Then ๐ด๐›ผ=๐›ผ(ฬƒ ๐œ‡๐‘—) and ๐ด๐›พ=๐›พ๐œ†ฬƒ .Thesemeanthat(๐œ‡๐‘—) and ๐œ† Phys.Theory,vol.16,p.157,2003. are two coneigenvalues of quaternion matrix ๐ด,andif๐›ฝ =0ฬธ , [15] S. de Leo and P. Rotelli, โ€œOdd-dimensional translation between then ๐›ผ and ๐›พ are corresponding coneigenvectors related to (๐œ‡๐‘—) complex and quaternionic quantum mechanics,โ€ Progress of โˆ’1 Theoretical Physics,vol.96,no.1,pp.247โ€“255,1996. and ๐œ†,respectively.Moreover๐‘ฬƒ ๐œ†๐‘ are right coneigenvalues of ๐ด,and๐›พ๐‘ฬƒ is corresponding coneigenvector for any nonzero [16] S. de Leo and G. Scolarici, โ€œRight eigenvalue equation in quater- nionic quantum mechanics,โ€ JournalofPhysicsA,vol.33,no.15, quaternion ๐‘. pp. 2971โ€“2995, 2000. [17] S. de Leo, G. Scolarici, and L. Solombrino, โ€œQuaternionic eigen- Conflict of Interests value problem,โ€ Journal of Mathematical Physics,vol.43,no.11, pp. 5815โ€“5829, 2002. The authors declare that there is no conflict of interests [18] H. Liping, โ€œConsimilarity of quaternion matrices and complex regarding the publication of this paper. matrices,โ€ Linear Algebra and Its Applications,vol.331,no.1โ€“3, pp.21โ€“30,2001. Acknowledgments [19] T. Jiang and S. Ling, โ€œAlgebraic methods for condiagonaliza- tion under consimilarity of quaternion matrices in quaternionic The authors would like to express their sincere thanks to the quantum mechanics,โ€ Advances in Applied Clifford Algebras,vol. referees for the careful reading and very helpful comments on 23,no.2,pp.405โ€“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).

References

[1] J. J. Sakurai, Modern Quantum Mechanics, Benjamin Cum- mings,MenloPark,Calif,USA,1985. [2] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge Uni- versity Press, New York, NY, USA, 1990. [3]Y.P.Hong,Consimilarity: theory and applications [doctoral dis- sertation], Johns Hopkins University, 1985. [4] Y.P.Hong and R. A. Horn, โ€œAcanonical form for matrices under consimilarity,โ€ Linear Algebra and Its Applications,vol.102,pp. 143โ€“168, 1988. Advances in Advances in Journal of Journal of Operations Research Decision Sciences Applied Mathematics Algebra Probability and Statistics Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014

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