Jordan Canonical Form of A Partitioned Complex Matrix and Its Application to Real Quaternion Matrices

COMMUNICATIONS IN ALGEBRA, 29(6, 2363-2375(200 Jordan Canonical Form of A Partitioned Complex Matrix and Its Application to Real Quaternion Matrices Fuzhen Zhang Department of Math Science and Technology Nova Southeastern University Fort Lauderdale, Florida 3334, USA E-mail: zhang@nova.edu Yimin Wei Department of Mathematics Fudan University Shanghai 200433, P. R. China E-mail: ymwei@fudan.edu.cn Abstract Let Σ be the collection of all 2n 2n partitioned complex matrices ( A A 2, A 2 A where A and A 2 are n n complex matrices, the bars on top of them mean matrix conjugate. We show that Σ is closed under similarity transformation to Jordan (canonical forms. Precisely, any matrix in Σ is similar to a matrix in the form J J Σ via an invertible matrix in Σ, where J is a Jordan form whose diagonal elements all have nonnegative imaginary parts. An application of this result gives the Jordan form of real quaternion matrices.. Isomorphism and complex adjoint of real quaternion matrices This research was motivated by studying the analogous properties of quaternion matrices to those of complex matrices. Throughout this paper we denote by C and Q, respectively, the sets of complex numbers and real quaternions and let F n and M n (F bethen-tuples and the n n

2 FUZHEN ZHANG matrices over F, wheref = C or Q. We call real quaternions simply quaternions, and take {, i, j, k} as a basis of Q. In addition, for A M n (Q, A denotes the conjugate of A and A = A T is the (quaternionic conjugate transpose of A. Let A M n (Q. Write A = A + A 2 j,wherea and A 2 are n n complex matrices. We associate with A the 2n 2n complex matrix ( A A 2 φ(a = ( A 2 A and call φ(a thecomplex adjoint matrix of the quaternion matrix A. LetΣbethe collection of all 2n 2n partitioned complex matrices in the form (. The mapping A φ(a is an isomorphism between M n (Q and Σ. Some basic properties of φ are (see [] φ(a + B =φ(a+φ(b; 2 φ(ab =φ(aφ(b; 3 φ(a =(φ(a ; 4 φ(a =(φ(a if A is invertible; 5 A has rank r if and only if φ(a hasrank2r; 6 φ(a is unitary, Hermitian, or normal if and only if A is unitary, Hermitian, or normal. Complex adjoint matrices have been employed in the study of quaternion matrices by many researchers (see, e.g., [4], [6], [9] or []. In order to obtain some properties such as commutativity, normality and polar decomposition for quaternion matrices, Wiegmann [8] investigated the Jordan form of a quaternion matrix through its adjoint using solely the complex matrix theory. He claimed the statement in our abstract. His proof, however, is false. The purpose of the present paper is to complete the proof. For a survey on quaternion matrices, see []. For other aspects of quaternions such as Cayley numbers and quaternionic quantum mechanics, see [7] and []. The problems we shall study in this paper are formulated as follows: Problem : Any A Σ has a Jordan form in Σ; Problem 2: Any A Σ is similar to a Jordan form in Σ via an invertible P Σ; Problem 3: Any A M n (Q has a Jordan form in M n (C in the usual sense.

JORDAN FORM OF A PARTITIONED MATRIX 3 Problems 2 and 3 are equivalent, since if P AP = B holds over Q, then (φ(p φ(aφ(p =φ(b holdsoverc. Conversely, if there exists a nonsingular matrix S in Σ such that S φ(as = T,thenT is in Σ. If we write ( S S S = 2 S 2 S ( T T, T = 2 T 2 T then the identity S φ(as = T or φ(as = ST implies It follows that A S A 2 S 2 = S T S 2 T 2 and A S 2 + A 2 S = S T 2 + S 2 T., (A + A 2 j(s + S 2 j=(s + S 2 j(t + T 2 j. Therefore the quaternion matrix A is similar to T +T 2 j via the nonsingular quaternion matrix S +S 2 j. We note that in such a way many quaternion matrix problems can be converted into partitioned complex matrix problems. Recall that the left and right eigenvalues λ of a square quaternion matrix A, defined respectively by Ax = λx and Ax = xλ for some nonzero quaternion vector x, are different in general. It has been evident that right eigenvalues, simply called eigenvalues, are more useful in the study of quaternion matrices. For left eigenvalues, one may refer to [0] (for the existence. Furthermore, any n n quaternion matrix has n complex right eigenvalues with nonnegative imaginary parts []. Problem is immediate from Problems 2 and 3, but not conversely, since the invertible matrix through which A is similar to its Jordan form in Σ need not be in Σ. Take, for instance, A = ( 0 0 Then the eigenvalues of A are i, i, anda is similar to its Jordan form J = ( i 0 0 i.. It is easy to verify that ( P AP = J, where P = i i Σ. However, as seen later, one can choose an invertible matrix in Σ that does the job. We give an affirmative answer to Problem 2, consequently to Problems and 3.

4 FUZHEN ZHANG 2. Adjoint vectors and lemmas As we saw in the previous section, an invertible matrix P that takes a matrix A Σ to its Jordan form need not be in Σ. Suppose such a P exists in Σ. If (v i,...,v ni,w i,...,w ni T is the i-th column vector of P, i n, then( w i,..., w ni, v i,...,v ni T should be the (n+i-th column vector of P. This vector is called the adjoint vector of the former one. In general, the adjoint vector v of a column vector v of 2n-complex components is defined to be ( v v2 = v, if v = ( v v 2, v,v 2 C n. Obviously v and v are linearly independent if v 0. In fact, they are orthogonal. It is readily seen that for any vectors u and v in C 2n and any complex number c (cu + v = cu + v, (u = u. (2 The following lemmas, extracted from [8], will be used in the proof of our main theorem when choosing basis vectors of the form v, v for certain eigenspaces. Lemma If v,v 2,...,v k C 2n are linearly independent, then so are v,v 2,...,v k. Lemma 2 Let v,v 2,...,v k,v k+ be vectors in C 2n. If v,v,..., v k,vk,v k+ are linearly independent, then v,v,...,v k,vk,v k+,vk+ are also linearly independent. Proof. Let a,b,...,a k+,b k+ be complex numbers such that a v + b v + + a kv k + b k v k + a k+v k+ + b k+ v k+ =0. Take adjoint ( to get a v b v + + a k v k b k v k + a k+ v k+ b k+ v k+ =0. Multiplying the first equation by a k+, and the second by b k+, then subtracting to get rid of vk+, one has a k+ 2 + b k+ 2 =0, since v,v,...,v k,v k,v k+ are linearly independent by assumption. Thus b k+ =0, hence all a s and b s equal zero due to the linear independence assumption.

JORDAN FORM OF A PARTITIONED MATRIX 5 Lemma 3 Let u C 2n, c C, anda, B Σ. Then [(A + cbu] =(A + cbu. (3 In particular [(A λiu] =(A λiu, (4 and if u is an eigenvector of A corresponding to λ, thensoisu to λ. 3. Jordan form of the partitioned matrices Jordan canonical forms play a fundamental role in linear algebra. The Jordan form of an n n complexmatrixcanbecarried out in three different classical ways: i using elementary divisor and invariant factor theory, ii by a pure matrix proof [3, p. 2], and iii utilizing the invariance of generalized eigenspaces ([5, p. 22]. We outline the last approach for convenience of the later use. The proof of our theorem is based on a thorough understanding and reconstruction of this approach. Step : Let λ C be an eigenvalue of A M n (C. The null spaces N((A λi k ={x C n (A λi k x =0}, k =, 2,..., form an increasing chain of subspaces of C n. Thus some two consecutive null spaces, and from there on, are identical. Let r be the smallest of such positive integers, i.e., N((A λi r N((A λi r =N((A λi r+. The subspace N((A λi r, abbreviated N λ, is the generalized eigenspace of A corresponding to λ, andr is the index of λ. Note that N λ is an invariant subspace of C n under A because (A λi i and A commute. For the same reason, the range R λ = {(A λi r v v C n } is also an invariant subspace under A. In addition, C n = N λ R λ. Step 2: Let λ,λ 2,...,λ k C be distinct eigenvalues of A. It can be shown that N λi R λj for any i j. This implies the key decomposition to the entire proof C n = N λ N λ2 N λk. (5 Since each N λi is an invariant subspace under A, one may focus on A on a typical generalized eigenspace N λ and show that the matrix of A under certain basis of N λ is of the desired Jordan form.

6 FUZHEN ZHANG Step 3: Let x N λ. Then (A λi r x =0. Ifx 0, then there exists an integer s r such that (A λi s x =0, but (A λi s x 0. Such a vector x is referred to as a generalized eigenvector of rank s corresponding to λ. A useful fact is that the generalized eigenvectors of different ranks corresponding to the eigenvalue λ are linearly independent. Step 4: Consider N λ with the index of λ being r, andlet N i = {x C n (A λi i x =0}, i =, 2,...,r. Then N N 2 N r = N λ. Note that the set N i+ N i contains all generalized eigenvectors of rank i +. We may form a basis for each N i+ by combining a set of basis vectors of N i and a set of linearly independent generalized eigenvectors of rank i +. LetT i be the subspace spanned by these generalized eigenvectors. Then N i+ = N i T i, i =, 2,...,r. With T 0 = N, we then have the decomposition for each N λ N λ = T r T r 2 T T 0. (6 Furthermore if vectors x,x 2,...,x p in T i are linearly independent, then the vectors (A λi j x, (A λi j x 2,..., (A λi j x p, j =, 2,...,i, (7 are also linearly independent. Step 5: We now choose a basis for N λ that gives the Jordan blocks of eigenvalue λ. Let the dimension of T i be d i and choose a basis for T i, i =0,,...,r, to form a basis by (6 and (7 for N λ as follows: T r : x,...,x dr ; T r 2 :(A λix,...,(a λix dr,x dr +,...,x dr 2 ; T r 3 :(A λi 2 x,...,(a λi 2 x dr, (A λix dr +,......,(A λix dr 2,x dr 2 +,...,x dr 3 ; T 0 :(A λi r x,...,(a λi r x dr, (A λi r 2 x dr +,......,(A λix d,x d +,...,x d0. Step 6: Rearrange the basis vectors just obtained. For x,wehave (A λi r x, (A λi r 2 x,..., (A λix, x.

JORDAN FORM OF A PARTITIONED MATRIX 7 Let P x be the n r matrix having these linearly independent vectors as columns. Then (A λip x = P x K, where K is the r r matrix with superdiagonal entries and 0 elsewhere. Thus AP x = P x J, (8 where J is the r r Jordan block with diagonal entries λ. Repeating the process for each x j in the basis of N λ, one obtains a Jordan form J λ which is a direct sum of the Jordan blocks of A belonging to λ through a matrix P λ with linearly independent column vectors. In symbols, AP λ = P λ J λ. Now let λ,λ 2,...,λ k be distinct eigenvalues of A M n (CandsetP =(P λ,p λ2,...,p λk. Then P is invertible and P AP is a Jordan form of A. For quaternion matrices, with Q n viewed as a right vector space over Q and with the (right eigenvalues of the matrices, this approach fails at the early stages in the Steps and 2; N λ is not an invariant subspace of A in general, since (A λi i and A do not commute if A is of quaternion entries. Thus the decomposition in (5 for C n does not hold for Q n in general. Aiming to resolve Problem 2 rather than dealing with the quaternion matrices directly, our strategy is to follow the above steps and show that any matrix in Σ has Jordan blocks in conjugate pairs. An invertible matrix P in Σ which gives the similarity A to its Jordan form is obtained by suitably choosing adjoint vectors to form bases for N λ and N λ if λ is nonreal, and for N λ itself if λ is real. To this end, we need to use the basic properties of the matrices in Σ. Notice that Σ is closed under addition, multiplication, and inversion; namely, A, B Σ A + B, AB, A Σ if the inverse exists. We observe that the nonreal eigenvalues of a matrix in Σ occur in conjugate pairs. A stronger result is the following statement ([2, p. 83]. Lemma 4 An n-square complex matrix A is similar to A or A if and only if the Jordan blocks of the nonreal eigenvalues of A occur in conjugate pairs. As a consequence, any matrix A Σ is similar to a Jordan canonical form J J R, where J has diagonal entries nonreal, and R is real. Note that if A has no real eigenvalues then Problem is settled and that A and A are not necessarily similar if only the occurrence of nonreal eigenvalues in conjugate pairs is assumed.

8 FUZHEN ZHANG Now that the eigenvalues of A Σ occur in conjugate pairs, we have the decomposition in the Step 3 for C 2n : C 2n = N λ N λ N λ2 N λ2 N λp N λp N λp+ N λq, (9 where λ s are distinct eigenvalues of A, λ,...,λ p nonreal and λ p+,...,λ q real. With (9 we shall proceed the Step 4 and consider N λ and N λ together if λ is nonreal. For the real case, we choose paired vectors v, v to form a basis for N λ. Theorem For any A Σ there exists an invertible matrix P Σ such that P AP = J J Σ is a Jordan canonical form of A, wherej has all its diagonal entries with nonnegative imaginary parts. Proof. Let λ be a nonreal eigenvalue of A. Thenλ is also a nonreal eigenvalue of A, andλ λ. We are to pair the generalized eigenspaces N λ and N λ. If the index of λ is r, then so is the index of λ, because, by Lemma 3, (A λi i x =0 (A λi i x =0. (0 If we denote N i and T i for λ as N i and T i for λ in the Step 4, then (6 and (7 hold side by side for N λ and N λ. And furthermore, the basis vectors for T i in the Step 5 that are used to form the basis N λ will produce, by Lemma 2 and with the replacement of λ by λ and of x by x, the basis vectors for T i to form the basis N λ. Since λ and λ are different eigenvalues of A and the decomposition (9 is a direct sum, all the vectors just obtained for N λ and N λ are linearly independent. Thus as the identity (8 in the Step 6, AP x = P x J and AP x = P x J both hold. Therefore A(P x,p x =(P x,p x (J J. The Jordan blocks J and J are thus paired. Note that the column vectors of (P x,p x are linearly independent. Repeat this process for every nonreal eigenvalue λ of A to pair all the Jordan blocks J and J of λ. Now we deal with the case in which λ is real. To carry out the Step 5, we first show that T r has a basis consisting of vectors in form v, v. Let 0 x T r be of rank r. Then the rank of x is also r by (0. Thus x T r,andx and x are linearly independent. If the dimension of T r is 2, we then turn to T r 2. Otherwise, let x 2 T r be linearly independent of x and

JORDAN FORM OF A PARTITIONED MATRIX 9 x. Then by Lemma 3, x,x,x 2,x 2 are linearly independent. Continuing this way, we have a basis for T r consisting of paired vectors x,x,...,x k,x k for some k. For T r 2, we apply A λi to the above vectors to get linearly independent vectors of rank r (A λix, (A λix,..., (A λix k, (A λix k. ( Note that each (A λix i is the adjoint of (A λix i. The vectors in ( are also in pairs v, v.ifx k+ is a vector of rank r which is not a linear combination of the vectors in (, then by Lemma 3 the vectors in ( plus x k+ and x k+ are linearly independent. In this way we obtain a basis of paired vectors v, v for T r 2. The same idea applies to all T i in the Step 5 to yield the basis vectors of N λ in pairs v =(A λi i x, v =(A λi i x (regard (A λi 0 = I. Now in the Step 6, we have AP x = P x J and AP x = P x J. It follows that A(P x,p x =(P x,p x (J J. The real Jordan blocks are thus paired. Note that the column vectors of (P x,p x are linearly independent. The same argument works as well for all real eigenvalues. We now put all the matrices P x and P x obtained from the above process for all eigenvalues λ, real and nonreal, to form an 2n 2n matrix P.ThenAP = P (J J (permutations may be applied if necessary. Matrix P is invertible, because its column vectors are the basis vectors of the subspaces in the direct sum decomposition (9. It is readily seen that P is in Σ. As a consequence, we have the following result. Theorem 2 Let A be an n n quaternion matrix with distinct (right eigenvalues λ,λ 2,...,λ s, s n, whose imaginary parts are nonnegative. Then there exists an invertible quaternion matrix P such that P AP = J J 2 J s is in a unique Jordan canonical form, where each J t is a direct sum of the Jordan blocks of λ t. The uniqueness follows immediately from that of the complex case. 4. Wiegmann s proof It has been evident that some quaternion matrix problems (such as the existence of left eigenvalues are much more difficult than expected (see [2] and [0]. It does not seem easy to give a direct and elementary proof for Theorem 2. Wiegmann

0 FUZHEN ZHANG studied Problem 2 using only the complex matrix theory. For A Σ, as a square complex matrix, A has a Jordan form J via an invertible matrix P. P and J need not be in Σ. The idea of Wiegmann s proof is to change the column vectors of P step by step to linearly independent vectors that span the same subspaces of certain column vectors of P, without altering the relations of the vectors to the partitioned matrix A, so that the resulting matrix P is of the desired form. But there is an invalid statement in his proof. To understand Wiegmann s idea and see what is wrong with the proof, we follow his line of proof, with instead of for the adjoint vectors, since the latter is now commonly used as conjugate transpose in matrix theory. In addition, our A Σis the same as A in [8]. Let A be a matrix in Σ. Let P be an invertible complex matrix such that P AP = J or AP = PJ, (2 where J is an ordinary Jordan form of A, i.e., the direct sum of Jordan blocks λ 0.......... 0 λ Thus each column vector v of P satisfies one and only one of the following relations: (i Av = λv or (ii Av = w + λv, where w is the column vector adjacent to v on the left. Let λ,λ 2,...,λ m be distinct eigenvalues of A, andletv λi be the vector space spanned by the column vectors of P corresponding to λ i, i =, 2,...,m.Then C 2n = V λ V λm. Since the above decomposition is a direct sum, one may focus on a typical subspace V λ. The strategy of Wiegmann s proof is to choose basis vectors of pairs v, v for V λ if λ is real and for V λ and V λ if λ is nonreal, which satisfy the relations (i and (ii. We shall see a false statement in Wiegmann s proof for the real case. Let λ be a real eigenvalue of A. Consider the column vectors of P that correspond to λ. Ifv denotes the first column vector of P for the first Jordan block of λ, write

JORDAN FORM OF A PARTITIONED MATRIX the remaining vectors of P for this block successively as v ( v v ( v (2 λ 0 0 λ 0 0 λ.,v(2,...,illustrated by Note that v is an eigenvector of A belonging to λ. Repeating for all other Jordan blocks of λ, we have the column vectors of P corresponding to the second, third,..., Jordanblocksofλ labeled respectively as v 2,v ( 2,v(2 2,..., v 3,v ( 3,v(2 3,...,... If we write SpanS to denote the space generated by the vectors in S, then V λ = Span{v,v 2,v 3,...} Span{v (,v( 2,v( 3,...} Span{v(2,v(2 2,v(2 3,...} (3 Note that the first summand Span{v,v 2,v 3,...} in (3 is the eigenspace of λ and it is invariant under A. Wiegmann s idea is to replace the basis vectors in each span by linearly independent vectors of the form v, v. Note that if v is a type (i vector for A, soisv by Lemma 3. One can show by using Lemma 2 that the vector space spanned by type (i vectors for λ is of even dimension and has a basis consisting of paired vectors in the form v, v. If v is a type (ii vector, then there is a nonzero vector w such that which, since λ is real, results in (A λiv = w (4 (A λiv = w. (5 Now let there be 2k column vectors v,v 2,...,v p,...,v 2k of P of type (i for λ, where p is the number of Jordan blocks of λ with size more than. Then (A λiv ( i = v i, i =, 2,...,p. It can be shown that p is even, for if v and v ( satisfy (5, then so do v and (v (. Moreover if v and v ( are linearly independent, then so are v and (v ( by Lemma. Either p = 2 or the process can be continued, so that p = 2q. It was concluded in Wiegmann s proof that in this way we see that there

2 FUZHEN ZHANG exists a set of linearly independent vectors v,v,...,v q,vq,...,...,v k,vk such that v (, (v(,...,v q (, (v q ( provide a basis for the space spanned by v (,...,v( 2q as taken above where v i and v ( i are related as above, and that the process can be repeated for v (2 j s, v (3 j s,...,until2nlinearly independent vectors which form amatrixp of the desired form are obtained. Putting this all in symbols, Span{v,v 2,v 3,...} = Span{v, v,...}, (6 Span{v (,v( 2,v( 3,...} = Span{v(, (v (,...}, (7 Span{v (2,v(2 2,v(2 3,...} = Span{v(2, (v (2,...}, We point out that the assertion that v (, (v(,...,v q (, (v q ( provide a basis for the space spanned by v (,...,v( 2q is false. That is, (7 and others, except (6, do not hold in general. Thus the process cannot be repeated. For a counterexample, take A = J = 2 0 0 0 2 0 0 0 0 2 0 0 0 2 and P = 0 0 0 0 0 0 Then AP = PJ. Notice that the first and the third column vectors of P are eigenvectors of A belonging to λ. Thus the column vectors of P from left to right are respectively v,v (,v 2,v ( 2. It is easy to check that v( and (v ( do not provide a basis for the space spanned by v ( and v ( 2, since (v( =(, 0,, T is not a linear combination of v ( and v ( 2. We conclude that Wiegmann s proof is false. ACKWLEDGEMENTS Work of the first author was supported in part by the Nova Faculty Development Funds. The work of the second author (Project 990006 was supported by the National Natural Science Foundation of China when he visited Harvard University..

JORDAN FORM OF A PARTITIONED MATRIX 3 References [] Adler, S. L. Quaternionic Quantum Mechanics and Quantum Fields. Oxford University Press: New York, 995. [2] Cohn, P. M. Skew Field Constructions. Cambridge University Press: London, 977. [3] Horn, R. A.; Johnson, C. R. Matrix Analysis. Cambridge University Press: New York, 985. [4] Huang, L. Jordan Canonical Form of a Matrix over the Quaternion Field. Northeast Math J. (China 994, 0 (, 8 24. [5] Lancaster, P.; Tismenetsky, M. The Theory of Matrices. Academic Press: Orlando, 985. [6] So, W.; Thompson, R. C.; Zhang, F. The Numerical Range of Normal Matrices with Quaternion Entries. Linear and Multilinear Algebra 994, 37, 75 95. [7] Ward, J. P. Quaternions and Cayley Numbers. Kluwer Academic Publishers: Norwell, 997. [8] Wiegmann, N. A. Some Theorems on Matrices with Real Quaternion Elements. Canad. J. Math. 995, 7, 9 20. [9] Wolf, L. A. Similarity of Matrices in Which the Elements Are Real Quaternions. Bull. Amer. Math. Soc. 936, 42, 737 743. [0] Wood, R. M. W. Quaternionic Eigenvalues. Bull. London Math. Soc. 984, 7, 37-38. [] Zhang, F. Quaternions and Matrices of Quaternions. Linear Algebra and Its Applications 997, 25, 2 57. [2] Zhang, F. Matrix Theory: Basic Results and Techniques. Springer: New York, 999.