# An Algebraic Relation between Consimilarity and Similarity of Quaternion Matrices and Applications

Authors: Tongsong Jiang; Xuehan Cheng; Sitao Ling
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

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.


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## Body

## 1. Introduction

An antilinear operatorT is a mapping from one complex vector space V into another W, which is additive and conjugate homogeneous, that is, for all α,β∈V and any complex number a, T(α+β)=T(α)+T(β), T(aα)=a¯T(α) in which a¯ is the conjugate of a. Two n×n complex matrices A,B are said to be consimilar if S-1AS¯=B for some n×n nonsingular complex matrix S. Consimilarity of complex matrices arises as a result of studying an antilinear operator referred to different bases in complex vector spaces, and the theory of consimilarity of complex matrices plays an important role in quantum mechanics [1, 2]. Complex consimilarity is an equivalent relation and has been studied [2–5].In recent years, applications of quaternion matrices are getting more and more important and extensive in quantum mechanics, rigid mechanics, and control theory [6–17]; it is becoming more and more necessary to study the theory and methods of quaternion matrices. In paper [18], the author introduced concepts of consimilarity of quaternion matrices. If A and B are both quaternion matrices of n×n, they are said to be consimilar if S-1AS~=B holds for some n×n nonsingular quaternion matrix S. Write A~sB if A is similar to B, A~csB if A is consimilar to B, and A~psB if A is permutation similar to B. Permutation similarity is both a similarity and consimilarity relations. A quaternion λ is said to be the right coneigenvalue attributed to A if the con-eigenequation Aα~=αλ holds. The consimilarity of quaternion matrices is natural extension of that of complex matrices and will have potential applications in the study of theory and numerical computations in modern quantum mechanics, and so forth.LetR be the real number field, C=R⊕Ri the complex number field, and H=R⊕Ri⊕Rj⊕Rk the quaternion field, where ij=-ji=k, i2=j2=k2=-1. Fm×n denotes the set of all m×n matrices on a field F. For A∈Hm×n, let AT be the transpose of A and A¯ the conjugate matrix of A. For a quaternion α=a1+a2i+a3j+a4k with at∈R, we use α¯=a1-a2i-a3j-a4k and α~=a1-a2i+a3j-a4k to denote the conjugate and j-conjugate of α, respectively. Let A=A1+A2i+A3j+A4k, where As∈Rm×n; define A~=-jAj=A1-A2i+A3j-A4k to be j-conjugate of A. It is easy to verify that jAj=-A~, A~~=A, and
(1)A+B~=A~+B~,AC~=A~C~,
for any two n×n matrices A,B and a n×n matrix C with quaternion entries. Obviously, A~=A¯ if A is a complex matrix.By means of real representation of complex matrices in paper [5], we studied the properties of consimilarity of complex matrices and gave a relation between consimilarity and similarity of complex matrices. This paper, by means of a complex representation of a quaternion matrix, studies the relation between consimilarity and similarity of quaternion matrices and derives an algebraic relation between consimilarity and similarity on quaternion field. Finally this paper gives an application on coneigenvalues and coneigenvectors of quaternion matrices.


## 2. Preliminaries

For a quaternion matrixA of dimension n×m, denote by A=B1+B2i+B3j+B4k≡A1+A2j, where A1=B1+B2i, A2=B3+B4i. The complex representation of the quaternion matrix A is defined [19] to be
(2)Aσ=[A2-A1A¯1A¯2]∈C2n×2m;
the complex matrix Aσ is known as complex representation of the quaternion matrix A. It is easy to verify that Aσ¯=(A~)σ.LetA,B∈Hn×m, C∈Hm×p, a∈R. Then by the definition of complex representation we easily get the following results:
(3)(A+B)σ=Aσ+Bσ,(aA)σ=aAσ;(4)QnTAσQm=Aσ¯=(A~)σ;(5)(AC)σ=AσQmCσ=Aσ(C~)σQp=Qn(A~)σCσ,
in which Qt=[0It-It0] with It being the identity matrix, and Qt2=-I2t, QtQtT=QtTQt=I2t. Clearly, from (5) we know that a quaternion matrix A is nonsingular if and only if complex representation matrix Aσ is nonsingular.ForA∈Hn×n and α∈C2n×1, if Aσα=λα, then by (4) we have
(6)A¯σ(QnTα)=λ(QnTα)⟺Aσ(QnTα¯)=λ¯(QnTα¯).
This means that the eigenvalues of complex representation Aσ appear in conjugate pairs.In the same manner, for a Jordan blockJr(λ) of an eigenvalue λ, if AσX=XJr(λ), then by (4) we have
(7)A¯σ(QnTX)=(QnTX)Jr(λ)⟺Aσ(QnTX¯)=(QnTX¯)Jr(λ¯).
And this means Jordan blocks Jr(λ) of complex representation Aσ appear in conjugate pairs.From the statement above we have the following result.Proposition 1.
LetA∈Hn×n. Then(1)
the real eigenvalues of complex representationAσ appear in pairs, and the imaginary eigenvalues of complex representation Aσ appear in conjugate pairs; that is, if complex eigenvalue λ is an eigenvalue of Aσ, then complex eigenvalue λ¯ is also an eigenvalue of Aσ,(2)
the real Jordan blocksJr(λ) of complex representation Aσ appear in pairs, and the imaginary Jordan blocks Jr(λ) of complex representation Aσ appear in conjugate pairs; that is, if Jr(λ) is a Jordan block of complex representation Aσ related to complex eigenvalue λ, then Jr(λ¯) is a Jordan block of complex representation Aσ related to complex eigenvalue λ¯.The following Proposition2 comes from the fact that jAj=-A~, (Jr(a+bi))j(i+j)~=(i+j)Jr(-b+aj) and ((A1+A2i)j)(i+j)~=(i+j)(-A2+A1j) by direct calculation for real numbers a and b, and real matrices A1 and A2.Proposition 2.
LetA be a quaternion matrix. Then(1)
Aj=jA~; this means that A is consimilar to A~; that is, A~csA~;(2)
ifJ=Jr(a+bi) is a Jordan block, a,b∈R, then Jj~csJr(-b+aj);(3)
ifA=A1+A2i is a complex matrix, A1,A2∈Rn×n, then
(8)Aj~cs-A2+A1j,Aj~cs(A1+A2j)j.Based on the quaternion fieldH, the author in [18] gave the following results.Proposition 3 (see [18] ).
IfA,B∈Hn×n, then
(9)A~csB⟺jA~sjB⟺Aj~sBj⟺jA~sBj.Proposition 4 (see [18] ).
IfA∈Hn×n, then
(10)A~csJn1(λ1)⊕Jn2(λ2)⊕⋯⊕Jnr(λr)≡JAc,
in which λt=at+btj are right coneigenvalues of A, at,bt are real numbers, and at≥0. JAc is uniquely determined by A up to the order of Jordan blocks Jnt(λt), and JAc is said to be the Jordan canonical form of A under consimilarity.


## 3. An Algebraic Relation between Consimilarity and Similarity

This section gives an algebraic relation between consimilarity and similarity of quaternion matrices by means of the complex representation in (2).LetA,B∈Hn×n be two quaternion matrices. If A is consimilar to B, then there exists a nonsingular quaternion matrix S such that AS=S~B, by (5) AσQnSσ=QnSσBσ. This means that if A is consimilar to B, then Aσ is similar to Bσ.Conversely, ifAσ is similar to Bσ, then Aσ and Bσ have the same eigenvalues. By Proposition 1, let λ1,λ¯1,…,λr,λ¯r be all eigenvalues and Jn1(λ1),Jn1(λ¯1),…,Jnr(λr),Jnr(λ¯r) corresponding Jordan blocks. There exists a complex and full-rank matrix Xj by [20, chapter 6.7] such that AσXt=XtJnt(λt), t=1,…,r. Then by (4), we have
(11)AσXt=XtJnt(λt),Aσ(QnTX¯t)=(QnTX¯t)Jnt(λ¯t),
and (11) is equivalent to
(12)Aσ(Xt,QnTX¯t)=(Xt,QnTX¯t)[Jnt(λt)00Jnt(λ¯t)].
Therefore there exists a nonsingular complex matrix T=(Z,QnTZ¯)∈C2n×2n, Z∈C2n×n such that
(13)T-1AσT=(Jj)σ=[∑t⊕Jnt(λt)00∑t⊕Jnt(λ¯t)],
in which J=∑t⊕Jnt(λt), λt=at+bti, at,bt are real numbers, and bt≥0.LetZ=[Z1Z2], Z1,Z2∈Cn×n, and S=Z¯2+Z1j. Then T=[Z1-Z¯2Z2Z¯1]=Sσ. From the nonsingular complex matrix T, we get S is a nonsingular quaternion matrix; by (1) and (5), (13) is equivalent to
(14)AσSσ=Sσ(Jj)σ⟺(AS~)σ=(S(Jj)~)σ⟺AS~=SJ~j=SJ¯j.
This means that A~csJ¯j. Let λt=at+bti, at,bt are real numbers, and bt≥0, and J¯=∑t⊕Jnt(λ¯t)=∑t⊕Jnt(at-bti). Then by Proposition 2 we have J¯j~cs∑t⊕Jnt(bt+atj), and we have A~cs∑t⊕Jnt(bt+atj). Similarly B~cs∑t⊕Jnt(bt+atj). Therefore A~csB.The statement above implies the following result.Theorem 5.
LetA,B∈Hn×n. Then A is consimilar to B if and only if complex representation matrix Aσ is similar tocomplex representation matrix Bσ; that is, A~csB if and only if Aσ~sBσ.Combining Proposition3 and Theorem 5, we get the following result.Corollary 6.
LetA,B∈Hn×n. Then
(15)A~csB⟺Aj~sBj⟺jA~sjB⟺jA~sBj⟺Aσ~Bσ.The proof of the Theorem5 is constructive, and the following results come from the proof above.Corollary 7.
LetA∈Hn×n. Then the following statements are equivalent.(1)
The Jordan canonical form of complex matrixAσ under similarity is
(16)J=∑t⊕Jnt(λt)⊕∑t⊕Jnt(λ¯t),
where λt=at+bti, at,bt∈R, bt≥0, t=1,…,r, are eigenvalues of Aσ.(2)
The Jordan canonical form of quaternion matrixA under consimilarity is
(17)Jc=∑t⊕Jnt(μt),
where μt=bt+atj, at,bt∈R, bt≥0, t=1,…,r, are coneigenvalues of A.Corollary 8.
LetA∈Hn×n. Then
(18)A~csJn1(μ1)⊕Jn2(μ2)⋯⊕Jnr(μr)≡Jc,
where μt=bt+atj=(1/2i)(λt-λ¯t)+(1/2)(λt+λ¯t)j are coneigenvalues of A, λt=at+bti, at,bt∈R, bt≥0, t=1,…,r, in which λ1,λ¯1,…,λr,λ¯r are all eigenvalues of complex representation Aσ, and Jn1(λ1),Jn1(λ¯1),…,Jnr(λr),Jnr(λ¯r) corresponding Jordan blocks in (16). Jc is uniquely determined by A up to the order of Jordan blocks Jnt(μt), and Jc is said to be the Jordan canonical form of A under consimilarity.Corollary 9.
LetA∈Hn×n. Then A has at least a coneigenvalue μt=Re(μt)+Im⁡(μt)j, Re(μt)≥0. If λ1,λ¯1,…,λr,λ¯r, λt=at+bti, at,bt∈R, bt≥0, t=1,…,r, are all eigenvalues of complex representation Aσ, then μ1,…,μr are coneigenvalues of quaternion matrix A, in which
(19)μt=bt+atj=12i(λt-λ¯t)+12(λt+λ¯t)j,
and for any nonzero quaternion p, p~-1μtp, t=1,…,r, are all coneigenvalues of quaternion matrix A.The following result comes immediately from Theorem5 and the definition of complex representation in (2).Corollary 10.
LetA,B∈Cn×n. Then A is consimilar to B if and only if [0-AA¯0] is similar to [0-BB¯0].


## 4. Algebraic Methods and Applications

This section gives an algebraic method for relation between consimilarity and similarity of quaternion matrices by means of complex representation and obtains an algebraic method for coneigenvalues and coneigenvectors for quaternion matrices as an application.LetA∈Hn×n, B∈Hm×m. Suppose there exists a complex matrix T∈C2n×2m such that AσT=TBσ. By (4)
(20)AσT=TBσ⟺Aσ(QnT¯QmT)=(QnT¯QmT)Bσ.
Let T^=(1/2)(T+QnT¯QmT). Then
(21)AσT^=T^Bσ.
Let
(22)T=[T11T12T21T22],
where Tkl∈Cn×m. It is easy to get by direct calculation
(23)T^=[T^1-T^2T^¯2T^¯1],
where
(24)T^1=12(T11+T¯22),T^2=12(T21-T¯12).
From (23) we construct a complex matrix S using the identities jT^t=T^t¯j and jT^tj=-T^t¯(25)S=T^2+T^1j=12(In,jIn)T^[Im-jIm]=14(In,jIn)(T+QnT¯QmT)[Im-jIm].
Clearly Sσ=T^. Therefore by (5) and jBj=-B~, (21) is equivalent to
(26)AσSσ=SσBσ⟺(AS~)σ=(SB~)σ⟺AS~=SB~⟺ASj~=SjB.The statement above implies the following result.Proposition 11.
LetA∈Hn×n, B∈Hm×m. Then there exists a quaternion matrix S∈Hn×m such that AS~=SB if and only if there exists a complex matrix T∈H2n×2m such that AσT=TBσ. In which case, there exists a complex matrix T such that AσT=TBσ; let
(27)S=14(In,jIn)(T+QnT¯QmT)[Im-jIm]j.
Then S is a quaternion matrix and AS~=SB.By Theorem5 and the statement above we have the following result.Corollary 12.
LetA,B∈Hn×n. Then A is consimilar to B if and only if Aσ is similar to Bσ; that is, A~csB if and only if Aσ~sBσ. In which case, there exists a nonsingular complex matrix T such that AσT=TBσ; let
(28)S=14(In,jIn)(T+QnT¯QnT)[In-jIn]j.
Then S is a quaternion matrix and AS~=SB, and if S is a nonsingular, then A~csB.Corollary12 gives an algebraic method for the relation between consimilarity and similarity of quaternion matrices by means of complex representation and turns theory of consimilarity of quaternion matrices into that of ordinary similarity of complex representation matrices.As a special case of Corollary12, we give an algebraic method for relation between coneigenvalues and coneigenvectors of a quaternion matrix and corresponding eigenvalues and eigenvectors of the complex representation matrix as follows.Corollary 13.
LetA∈Hn×n, λ∈H. Then there exists a vector α∈Hn×1 such that Aα~=αλ if and only if there exists a vector β∈H2n×1 such that Aσβ=βλσ. In which case, there exists a vector β∈H2n×1 such that Aσβ=βλσ; let
(29)α=14(In,jIn)(β+Qnβ¯Q1T)[1-j]j.
Then Aα~=αλ.Now we give algebraic methods for finding coneigenvalues and coneigenvectors of a quaternion matrix by means of complex representation.ForA∈Hn×n, if μ is a complex eigenvalue of Aσ with Aσβ=βμ, β∈C2n×1, β≠0, then by (11) and (12), Aσβ=βμ, Aσ(QnTβ¯)=(QnTβ¯)μ¯, and
(30)Aσ(β,QnTβ¯)=(β,QnTβ¯)[μ00μ¯]⟺AσT=T(μj)σ,
in which T=(β,QnTβ¯)∈C2n×2. By Corollary 13, let
(31)α=14(In,jIn)(β+Qnβ¯Q1T)[1-j]j.
Then Aα~=α(μ¯j). This means that μ¯j is a coneigenvalue of A and if α≠0, then α is a coneigenvector related to coneigenvalue μ¯j.By the statement above we get the following result.Corollary 14.
LetA∈Hn×n. If μ is a complex eigenvalue of complex representation Aσ and β is eigenvector related to μ with Aσβ=βμ, let
(32)α=14(In,jIn)(β+Qnβ¯Q1T)[1-j]j.
Then Aα~=α(μ¯j). This means if α≠0, then μ¯j is a coneigenvalue of A and α is a coneigenvector related to coneigenvalue μ¯j.Moreover, letμ=a+bi, a,b∈R, and b≥0. Then by Proposition 2 and the fact that ((a+bi)j)(i+j)~=(i+j)(-b+aj) we have
(33)Aβ~=β(μ¯j)⟺A(β(i+j)~)=(β(i+j))(b+aj)⟺Aγ~=γλ,
in which λ=b+aj and γ=β(i+j). That is λ is a coneigenvalue of quaternion matrix A, and γ is a coneigenvector related to λ.By Corollary14 and the statement above we have the following result.Theorem 15.
LetA∈Hn×n. If μ is a complex eigenvalue of complex representation Aσ and β is an eigenvector related to μ with Aσβ=βμ, let
(34)λ=12i(μ-μ¯)+12(μ+μ¯)j,α=14(In,jIn)(β+Qnβ¯Q1T)[1-j]j,γ=α(i+j)=14(In,jIn)(β+Qnβ¯Q1T)[1-j]j(i+j).
Then Aα~=α(μ¯j) and Aγ~=γλ. These mean that (μ¯j) and λ are two coneigenvalues of quaternion matrix A, and if β≠0, then α and γ are corresponding coneigenvectors related to (μ¯j) and λ, respectively. Moreover p~-1λp are right coneigenvalues of A, and γp~ is corresponding coneigenvector for any nonzero quaternion p.



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