Skew-coninvolutory matrices

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1 Linear Algebra and its Applications 426 (2007) Skew-coninvolutory matrices Ma. Nerissa M. Abara a, Dennis I. Merino b,, Agnes T. Paras a a Department of Mathematics, University of the Philippines, Diliman, Quezon City 0, Philippines b Department of Mathematics, Southeastern Louisiana University, Hammond, LA , United States Received 6 February 2007; accepted 24 May 2007 Available online 5 June 2007 Submitted by R.A. Brualdi Abstract We study the properties of skew-coninvolutory (EE = I) matrices, and derive canonical forms and a singular value decomposition. We study the matrix function ψ S (A) = SA S, defined on nonsingular matrices and with S satisfying SS = I or SS = I. We show that every square nonsingular A may be written as A = XY with ψ S (X) = X and ψ S (Y ) = Y. We also give necessary and sufficient conditions on when a nonsingular matrix may be written as a product of a coninvolutory matrix and a skew-coninvolutory matrix or a product of two skew-coninvolutory matrices. Moreover, when A is similar to A, or when A is similar to A, or when A is similar to A, or when A is similar to A, we determine the possible Jordan canonical forms of A for which the similarity matrix may be taken to be skew-coninvolutory Elsevier Inc. All rights reserved. AMS classification: 5A2; 5A23 Keywords: Coninvolutory matrices; Skew-coinvolutory matrices; Canonical forms. Introduction and notation A square complex matrix E is called coninvolutory if EE = I, that is, E is nonsingular and E = E. It is known [2,3 that every coninvolutory matrix E can be written as E = e ir for some real matrix R. Moreover, since E = E, then the singular values of E are either, or pairs of σ and /σ, where σ>. Corresponding author. addresses: issa@up.edu.ph (Ma.N.M. Abara), dmerino@selu.edu (D.I. Merino), agnes@math.upd.edu.ph (A.T. Paras) /$ - see front matter ( 2007 Elsevier Inc. All rights reserved. doi:0.06/j.laa

2 Ma.N.M. Abara et al. / Linear Algebra and its Applications 426 (2007) We consider the set of matrices A satisfying A = A. We call such matrix skew-coninvolutory. We derive properties and canonical forms for skew-coninvolutory matrices, analogous to known results for coninvolutory matrices. We let M n be the set of n-by-n complex matrices. In [4, a linear operator φ S on M n was defined by φ S (A) = SA T S, where S is either symmetric or skew-symmetric. It was shown that every nonsingular matrix A M n may be written as A = XY, where φ S (X) = X and φ S (Y ) = Y. Notice that when S = I, this is the classical algebraic polar decomposition of A. We consider the analogous function on the set of all nonsingular matrices in M n defined by ψ S (A) = SA S for some nonsingular S, and show that if we put the restriction ψ S (ψ S (A)) = A for all nonsingular A M n, then S may be chosen to be either coninvolutory or skew-coninvolutory. We give some properties of ψ S and prove a ψ S -polar decomposition for nonsingular matrices. We also determine the respective Jordan canonical forms of ψ S -orthogonal, ψ S -symmetric and ψ S -skew-symmetric matrices. 2. Properties and canonical forms Definition. Let n be a positive integer. We denote the set of skew-coninvolutory matrices by D n {A M n : AA = I}, and we denote the set of coninvolutory matrices in M n by C n. We also set E n C n D n. Notice that D n is empty when n is odd since det(aa) [ is nonnegative for any A M n. When n is even, say n = 2k, then D n is nonempty as J 0 Ik I k 0 D n. The following can be verified easily. Proposition 2. Let A M n be given. Any two of the following implies the third: (a) A is unitary. (b) A is skew-symmetric. (c) A is skew-coninvolutory. Given A D n and any nonsingular X M n, notice that (XAX )(XAX ) = XAAX = I n. That is, the set D n is closed under consimilarity and in particular, real similarity. Proposition 3. Let A D n be given. Then XAX D n for any nonsingular X M n. In particular, RAR and EAE are skew-coninvolutory matrices for real nonsingular R and coninvolutory E. It is natural to ask if two similar skew-coninvolutory matrices are also real similar (similar via a real matrix). Recall that two matrices A and B M n are real similar if and only if there exists a nonsingular S M n such that A = SBS and A = SBS [3, Theorem.. Proposition 4. Two matrices A, B D n are similar if and only if they are similar via a real matrix.

3 542 Ma.N.M. Abara et al. / Linear Algebra and its Applications 426 (2007) Proof. Since real similarity implies similarity, it suffices to prove necessity. Suppose A, B D n and suppose A = SBS. Then A = A = SB S = SBS. Let E D n be given. Then E = E. Thus, the Jordan blocks and singular values of E come in special pairs. Proposition 5. Let E D n be given. ( ) (a) If J k () is a Jordan block of E with multiplicity l, then J k is a Jordan block of E with multiplicity l. (b) If σ>0is a singular value of E with multiplicity l, then σ is a singular value of E with multiplicity l. (c) If σ>0and l is a positive integer, then there is a skew-coninvolutory F M 2l such that σ and σ are singular values of F, each with multiplicity l. [ 0 σil Proof. For (c), let σ>0be given. Notice that F = σ M I l 0 2l is a skew-coninvolutory matrix with σ and σ as singular values, each with multiplicity l. Let A M n be nonsingular. Suppose B = XAX is coninvolutory. Then B = B, so that XAX = XA X and A = X (XA X )X = (X X)A (X X). Notice that S X X is coninvolutory. Thus, if A is similar to a coninvolutory matrix, then A is similar to A via a coninvolutory matrix. One checks that the converse holds as well: if A is similar to A via a coninvolutory matrix, then A is similar to a coninvolutory matrix. Moreover, the same can be said when we replace coninvolutory with skew-coninvolutory. Proposition 6. Let A M n be nonsingular. Then (i) Ais similar to a coninvolutory matrix if and only if there exists S C n such that A = S(A )S ; and (ii) if n is even,ais similar to a skew-coninvolutory matrix if and only if there exists S C n such that A = S( A )S. Let E D n be given. ( Then ) Proposition 5 guarantees that if J k () is a Jordan block of E with multiplicity l, then J k is a Jordan block of E with multiplicity l. Since, /= for any C, we expect E to be similar to a matrix of the form A A. Proposition 7. A matrix E M 2n is similar to a skew-coninvolutory if and only if E is similar to A A for some nonsingular A M n. Proof. Let B = A A ( ) and S = 0 In I n 0. Then S is coninvolutory and B = S( B )S. By Proposition 6, B is similar to some E D 2n. Thus, if X is similar to B = A A, then X is similar to a skew-coninvolutory.

4 Ma.N.M. Abara et al. / Linear Algebra and its Applications 426 (2007) Conversely, suppose E( D) 2n and let J be the Jordan canonical form of E. If J k () is a Jordan block of E, then so is J k. Since cannot be equal to for all 0 /= C, then J may be written as )) J = m i= (J ki ( i ) J ki ( i, where m i= k i = n. Let A = m i= J k i ( i ). Then A = m i= J k i ( i ) which is similar to m i= J k i ( i ). Hence there exists a nonsingular X M n such that J = A X( A )X = (I n X)(A A )(I n X). Therefore E is similar to A A. The following matrix, defined in [3, was used to obtain examples of, and canonical forms for, coninvolutory matrices. Let k be a positive integer, let A, B M k, and define C 2k (A, B) [ A + B i(a B). 2 i(a B) A + B For 0 /= C,weletD 2k () E 2k (J k (), J k () ). It is known [ that C 2k (A, B) is similar to diag(a, B) via the unitary, symmetric and coninvolutory U = I ii. 2 ii I Lemma 8. Let A, B M k AB = I. be given. Then E 2k (A, B) is skew-coninvolutory if and only if Proof. Computations show that E 2k (A, B)E 2k (A, B) = [ AB + BA i(ab BA), () 2 i(ab BA) AB + BA hence, the lemma follows. We use the preceding lemma to obtain a canonical form under real similarity for skew-coninvolutory matrices. Theorem 9. Let E D n. Then there exist positive integers m, k,,k m, and scalars,..., m with i for each i =,...,msuch that 2 m i= k i = n and E is real similar to m i= D 2k i ( i ). Proof. Suppose E D n. By Proposition 5, we have that E is similar to J = m i=( Jki ( i ) J ki ( i ) ). By Lemma (g) of [3, J is similar to m i= D 2k i ( i ), which is skew-coninvolutory by Lemma 8. Hence, Proposition 4 guarantees that E is real similar to m i= D 2k i ( i ). Note that the matrix m i= D 2k i ( i ) is determined by the Jordan canonical form of E. Given a skew-coninvolutory matrix A M 2n, notice that A 2 is a coninvolutory matrix. From [3, we know that a coninvolutory matrix has Jordan blocks of the form ( ) (i) J k () J k, where /= 0, or (ii) J k (e iθ ), where θ R.

5 544 Ma.N.M. Abara et al. / Linear Algebra and its Applications 426 (2007) Among the coninvolutory matrices, it is natural ( to) ask which ones have a skew-coninvolutory square root. Note that if /= 0, then J k () J k is similar to J k () J k (), which is similar to a skew-coninvolutory matrix using Proposition 7. Theorem 0. Let A C 2n be given. Then A has a skew-coninvolutory ( ) square root if and only if its Jordan blocks may be arranged in the form J k () J k. Proof. Suppose A C 2n. Let A have Jordan canonical form ( )) J = m i= (J ki ( i ) J ki. i We choose K m i= (J ki ( ( i ) J ki )) i, which is similar to a skew-coninvolutory matrix by Proposition 7, that is, K = XMX for some nonsingular X M 2n and a skewconinvolutory M M 2n. Then K 2 = XM 2 X is similar to J, which implies A is similar to M 2. Since A and M 2 are both coninvolutory, then by Proposition 4, there exists a real matrix R M 2n such that A = RM 2 R. Since M is skew-coninvolutory and R is real, then by Proposition 3, RMR is skew-coninvolutory. Hence, A = (RMR ) 2 has a skew-coninvolutory square root. Conversely, suppose A = B 2 for a skew-coninvolutory B. Then B will have Jordan blocks of the form J k () J k ( ), where /= 0. Hence B 2 will have Jordan blocks of the form J k ( 2 ) J k ( 2 ) for /= 0. [ Let r be a positive integer and set J 2r 0 Ir I r 0. Note that J 2r is skew-coninvolutory. Moreover, J 2r = J2r T = J 2r. We use this to derive a singular value decomposition for skewconinvolutory matrices. Lemma. Let X M 2r be unitary and skew-symmetric, and let Y XJ 2r. Then Y is unitary and X = YJ 2r = J 2r Y T. Proof. Since X and J 2r are unitary, then so is Y = XJ T 2r. Moreover, X and J 2r are skew-symmetric and J 2r = J 2r, hence, YJ 2r = X = X T = J T 2r Y T = J 2r Y T. Given a skew-coninvolutory matrix E, we wish to find a singular value decomposition of the form E = W ΨW T, where W is unitary and Ψ is of some special form. First, consider a singular value decomposition E = UΣV of a skew-coninvolutory matrix E M n. By Proposition 5, Σ has the form Σ = (σ I n σ I n ) (σ k I nk σk I nk ) I nk+, (2) with σ >σ 2 > >σ k > and k i= 2n i = n n k+. We partition the unitary matrix X X 2 X l X 2 X 22 X 2l VU = X l X l2 X ll (3)

6 Ma.N.M. Abara et al. / Linear Algebra and its Applications 426 (2007) conformal to Σ, with X,X 22 M n,...,x l 2,l 2,X l,l M nk,x ll M nk+ and l = 2k +. Since E = E, then VUΣ = Σ (VU) T and following the argument in Theorem 5 of [3, we get [ [ 0 X 0 Xk VU = X T 0 Xk T X 0 k+, (4) where X k+ is unitary and skew-symmetric, [ and of even dimension n k+. 0 [ For i k, weletψ i = σi I ni Xi 0 and Y σ i I ni 0 i = 0 X i T [ [ 0 Xi σi I ni 0 Y i Ψ i = Xi T = Ψ 0 0 i Yi T. σ i I ni. Then For i = k +, Lemma guarantees that there exists a unitary Y k+ such that Y k+ Ψ k+ = Ψ k+ Yk+ T, where Ψ k+ J 2r and 2r = n k+. Now, Y i is unitary for i =,...,k,k+, hence, we can find a unitary and polynomial square root for Y i, say Z i. Since Y i Ψ i = Ψ i Yi T, then Z iψ i = Ψ i Zi T, for i =,...,k+. Let Y Y Y k Y k+, let Z Z Z k Z k+, and let Ψ Ψ Ψ k Ψ k+. Then VUΣ = Y Ψ = Z 2 Ψ = Z(ZΨ) = ZΨZ T. Hence, E = UΣV = V T (VUΣ)V = (V T Z)Ψ (V T Z) T = W ΨW T, where W = V T Z is unitary and Ψ = [ 0 σ I n σ I n 0 [ 0 σk I nk σ k I nk 0 [ 0 Ir. (5) I r 0 Conversely, if Ψ is of the form described in (5) which is skew-coninvolutory, and if W is unitary, then by Proposition 3, W ΨW T = W ΨW is skew-coninvolutory. Theorem 2. Let E D 2n be given. Then there exists a unitary W M 2n and ([ Ψ = k 0 ) [ σi I ni 0 Ink+ i=, (6) σ i I ni 0 I nk+ 0 with σ > >σ k > and k+ i= n i = n such that E = WΨW T. Conversely WΨW T is skewconinvolutory whenever W is unitary and Ψ is of the form (6). Since J 2k is skew-coninvolutory, then by Proposition 3, XJX is skew-coninvolutory for all nonsingular X M n. Let n, m be positive integers. Observe that J 2n J 2m is similar to J 2(n+m) via a permutation matrix. We use this to show that every skew-coninvolutory matrix is consimilar to J 2n for some n. Theorem 3. Let E M 2n be given. Then E is skew-coninvolutory if and only if E = XJ 2n X, for some nonsingular X M 2n. Proof. Note that for σ>0and a positive integer m, [ [ 0 [ [ σ I m 0 = σ I m 0 Im 0 σ I m. σi m 0 σim 0 I m 0 σim 0 Suppose E is skew-coninvolutory. Then by Theorem 2, there exists[ a unitary U and a skewconinvolutory Ψ of the form (6) such that E = UΨU T 0. Let W i σi I ni, for i = σi I ni 0

7 546 Ma.N.M. Abara et al. / Linear Algebra and its Applications 426 (2007) [ 0 Ink+,...,k, W k+ I nk+ 0 and W k+ i= W i. Then E = UΨU T = (UW)K(UW), where K = k+ i= J 2n i is similar to J 2n via a permutation matrix P. Hence, E = (UWP )J 2n (UWP). 3. The function ψ S (A) = SA S 3.. Properties of ψ S Let Mn be the set of nonsingular n-by-n matrices, and let S M n be given. We define ψ S : Mn M n by ψ S(A) = SA S. A similar function (φ S (A) = SA T S defined on M n )was used in [4 to further study the QS decomposition (orthogonal-symmetric) of a square matrix. We begin with the following. Lemma 4. Let S Mn be given. Then (a) ψ S (I) = I. (b) ψ S (AB) = ψ S (B)ψ S (A) for any A, B M n. (c) ψ S (A ) = ψ S (A) for any A M n. It is known [4 that if φ S (A) = SA T S satisfies φ S (φ S (A)) = A for all A M n, then S must be either symmetric or skew-symmetric. When S E n, then SS =±I so that ψ S (ψ S (A)) = ψ S (SA S ) = SSA(SS) = A. We now show that if the function ψ S (A) = SA S satisfies ψ S (ψ S (A)) = A for all A M n then S may be taken to be an element of E n. Proposition 5. Let X Mn be given. If ψ X(ψ X (A)) = A for all A Mn, then there exists S E n such that ψ X = ψ S. Conversely, if S E n, then ψ S (ψ S (A)) = A for all A Mn. Proof. If ψ X (ψ X (A)) = A for all A Mn, then XXA = AXX for all nonsingular A, and thus, XX = αi for some nonzero α C since X is nonsingular. Now, X = αx = α(αx ) = αα X. Therefore, αα =, that is, α R. Set S α /2 X and notice that ψ X (A) = ( α /2 S)A ( α /2 S ) = SA S = ψ S (A). Moreover, SS = α XX = α { I, if α>0, α I = I, if α<0. Hence S E n, as desired. The following definitions are analogs of the terminologies defined in [4. Definition 6. Let S E n be given. We say that A M n is ψ S-symmetric if ψ S (A) = A; A is called ψ S -orthogonal if ψ S (A) = A ; and A is called ψ S -skew-symmetric if ψ S (A) = A. Let S E n be given. Then ψ S (ψ S (A)) = A for all A M n. Because ψ S(AB) = ψ S (B)ψ S (A) for any A, B M n, as well, then ψ S(Aψ S (A)) = Aψ S (A). That is, Aψ S (A) is ψ S -symmetric for any A M n.

8 Ma.N.M. Abara et al. / Linear Algebra and its Applications 426 (2007) Lemma 7. Let S E n be given. (a) Aψ S (A) and ψ S (A)A are ψ S -symmetric for any A Mn. (b) If A and B are ψ S -orthogonal, then AB is ψ S -orthogonal. (c) Suppose S C n and suppose that S 2 = S with S E n. Then S AS is ψ S -symmetric if and only if A C n, and S AS is ψ S -skew-symmetric if and only if A D n. Proof. Claims (a) and (b) follow directly from Lemma 4. For (c), suppose S C n and suppose that S 2 = S with S E n. Then S = SS =±SS. Now, ψ S (S AS ) = S(S A S )S = (±S )A (±S ) = S A S. Notice that ψ S (S AS ) is equal to S AS if and only if A = A; and ψ S (S AS ) is equal to (S AS ) if and only if A = A ψ S -Skew-symmetric and ψ S -symmetric matrices Suppose that A M n is nonsingular and that A is similar to A. We show that for such a matrix A, the matrix of similarity may be chosen to be coninvolutory or skew-coninvolutory, that is, A = S( A )S, with S E n. Note that in this case, ψ S (A) = S(A )S = A, so that A is ψ S -skew-symmetric. Theorem 8. Let A M n be nonsingular. The following are equivalent: (a) A is similar to A. (b) A is similar to a skew-coninvolutory matrix. (c) A is similar to a skew-coninvolutory via a coninvolutory. (d) A is similar to A via a coninvolutory. (e) A is a product of a coninvolutory and a skew-coninvolutory. (f) A is similar to A via a skew-coninvolutory. Proof. Suppose A is similar to A (. Then A has Jordan canonical form J = J ki ( i ) J ki ( i )). Set B J ki ( i ) and note that A is similar to B B, so that by Proposition 7, A is similar to a skew-coninvolutory matrix. Suppose A = X BX, where B D n. Write X = RE, where R is real and E is coninvolutory. Then A = E R BRE, and by Proposition 3, R BR is skew-coninvolutory. Hence, A is similar to a skew-coninvolutory via a coninvolutory. Suppose A = E CE, where E C n and C D n. Then A = ECE, so that C = E ( A )E. Hence, A = E 2 ( A )E 2, and note that E 2 is a square of a coninvolutory matrix and hence, is also coninvolutory.

9 548 Ma.N.M. Abara et al. / Linear Algebra and its Applications 426 (2007) Suppose A = E ( A )E, where E is coninvolutory. Let Z = E ( A ) = AE, and observe that ZZ = E ( A )(AE ) = I. Hence, Z is skew-coninvolutory. Moreover, A = ZE is a product of a skew-coninvolutory matrix and a coninvolutory matrix. Suppose A = XY, where X is skew-coninvolutory and Y is coninvolutory. Then A = Y X = YX, which implies that Y = A X. Therefore, A = X( A )X for a skewconinvolutory X. One checks that (f) implies (a). We now look at a matrix A that is similar to A. We show A = SA S for some coninvolutory S so that A is ψ S -symmetric. Theorem 9. Let A M n be nonsingular. The following are equivalent: (a) A is similar to A. (b) A is similar to a coninvolutory matrix. (c) A is similar to a coninvolutory via a coninvolutory matrix. (d) A is similar to A via a coninvolutory matrix. (e) A is a product of two coninvolutory matrices. Proof. Suppose A is similar to( A ). If A has Jordan canonical form J, then J is a direct sum of blocks of the form J k () J k and J k (e iθ ) for θ R. Thus, J is similar to a coninvolutory matrix and therefore, so is A. Suppose A = X BX for some nonsingular X M n and a coninvolutory B M n. Write X = RE, where R is real and E is coninvolutory E. Then A = E R BRE and notice that R BR is also coninvolutory. Therefore A is similar to a coninvolutory via a coninvolutory. If A = X EX for coninvolutory matrices X and E, then A = XEX, so that E = X A X and A = (X 2 ) A X 2. Note that X 2 is coninvolutory. If A = E A E for a coninvolutory E, then E A = AE is coninvolutory. Therefore A is a product of two coninvolutories. If A = XY for coninvolutory matrices X and Y, then A = (XY) = Y X = YX. Hence Y = A X and A = XA X, that is, A is similar to A. The following theorem gives a necessary and sufficient condition for A to be similar to A via a skew-coninvolutory matrix. Theorem 20. Let A M 2n be given. Then A is similar to A via a skew-coninvolutory matrix if and only if A is a product of two skew-coninvolutory matrices. Proof. Suppose A = XY for skew-coninvolutory matrices X, Y. Then A = Y X = YX, hence Y = A X. Therefore, A = XA X, where X is skew-coninvolutory. Conversely,

10 Ma.N.M. Abara et al. / Linear Algebra and its Applications 426 (2007) suppose A = XA X for a skew-coninvolutory X, and observe that (A X )(A X ) = I. Hence, A is a product of two skew-coninvolutories ψ S -Polar decomposition Every nonsingular matrix A may be written as A = XY, where X is orthogonal and Y is symmetric. We now show that every nonsingular matrix A may be written as A = XY, where X is ψ S -orthogonal and Y is ψ S -symmetric. We make use of the following result that shows that every ψ S -symmetric matrix has a square root that is also ψ S -symmetric. Lemma 2. Let S E n be given and let A M n be ψ S -symmetric. Then there exists a ψ S - symmetric B M n such that B 2 = A. Proof. Suppose A is ψ S -symmetric. Then A is similar to A and, by Theorem 9, there exist matrices X, E C n such that A = XEX. Furthermore, Theorem.4 in [3 guarantees that E has a polynomial square root F, that is, F 2 = E and F = p(e) for some polynomial p(t). Set B XFX so that B 2 = A. Since A is ψ S -symmetric, we have XEX = S(XEX ) S = SX EXS, so that Xp(E)X = SX p(e)xs, that is, B = XFX = SX FXS = S(XF X ) S = ψ S (B), that is, B is a ψ S -symmetric as desired. Let A M n be nonsingular. Then A may be written as A = RE, where R is real and E is coninvolutory. Let S E n be given. Theorem 9 and Lemma 2 imply that ψ S (A)A = Y 2, with Y a ψ S -symmetric matrix. Define X AY. Then ψ S (X) = ψ S (AY ) = ψ S (Y ) ψ S (A) = Y ψ S (A) = Y (Y 2 A ) = YA = X. Thus, X is ψ S -orthogonal, Y is ψ S -symmetric and A = XY. We have proven the following. Theorem 22. Let A M n be nonsingular and let S E n. Then there exist X, Y M n such that X is ψ S -orthogonal, Y is ψ S -symmetric and A = XY. 4. Jordan canonical forms Let A M n be nonsingular. If A is similar to A, then Theorem 8 guarantees that the matrix of similarity may be chosen to be coninvolutory or skew-coninvolutory. If A is similar to A, then Theorem 9 ensures that the matrix of similarity may be taken to be coninvolutory. We

11 550 Ma.N.M. Abara et al. / Linear Algebra and its Applications 426 (2007) consider three classes of matrices: A similar to A, A similar to A and A similar to A. In all three cases, we prove that the matrix of similarity may always be taken to be coninvolutory. However, in each case, we show that the matrix of similarity may be chosen to be skew-coninvolutory only when the Jordan blocks of A occur in pairs with a particular form. One key observation is that an upper triangular matrix cannot be skew-coninvolutory. This is because if X is upper triangular, then the diagonal entries of the product XX are of the form x ii 2 for some x ii C, and thus cannot be equal to. The matrix J 2 () is similar to J J 2 () and J 2 J 2 () via the coninvolutory matrices diag(, ) and I 2, respectively. If X M 2 satisfies J 2 () = XJ k X, for k =, 2, then computations show that X must be upper triangular, and thus cannot be skew-coninvolutory. Therefore, J 2 () cannot be similar to its conjugate-inverse nor to its conjugate via a skew-coninvolutory matrix. Similarly, the matrix J 2 (i) is similar to J 2 (i) via the coninvolutory matrix diag(, ). Again, the matrix of similarity here cannot be chosen to be skew-coninvolutory. 4.. A similar to A The following lemma is easily verified and gives the result of conjugating a matrix A M 2k by the skew-coninvolutory matrix J 2k =. [ 0 Ik I k 0 Lemma 23. Let A =[A ij M 2k, where A ij M k and i, j =, 2. Then [ J 2k AJ 2k = A22 A 2 = J A 2 A 2k AJ 2k. We wish to characterize the matrices A which are similar to A via a skew-coninvolutory matrix by determining their Jordan canonical forms. We first give a class of matrices satisfying Theorem 20. Theorem 24. Let A M n have Jordan canonical form ( ( )) J = J k () J k. Then A is similar to A via a skew-coninvolutory matrix. ( ) Proof. If all the blocks in the Jordan canonical form of A occur in pairs J k () J k, then A is similar to J = (J k () J k () ), that is, A = PJP for some nonsingular P. By Lemma 23, J k () J k () = J 2k (J k () J k ())J 2k = J 2k (J k () J k () ) J 2k. Hence J = XJ X, where X = J 2k is skew-coninvolutory. Since A = PJP, then A = (P XP )A (P XP ). By Proposition 3, PXP is skew-coninvolutory, and thus A is similar to A via a skew-coninvolutory matrix.

12 Ma.N.M. Abara et al. / Linear Algebra and its Applications 426 (2007) A matrix that satisfies Theorem 20 satisfies Theorem 9. But a matrix of even order that is similar to its conjugate-inverse need not satisfy Theorem 20. Theorem 24 shows that a sufficient condition for a matrix A to be similar to A via a skew-coninvolutory matrix is to have its Jordan blocks occur in pairs J k () J k ( ). To show the necessity of this condition, we consider a Jordan matrix J similar to J. Suppose E = S JS such that E = X E X for some skewconinvolutory X. Then S JS = X S J SX, which implies J = (SXS ) J (SXS ), where SXS is skew-coninvolutory. Conversely, if E = SJS is the Jordan canonical form of E and if J = Y J Y for skew-coninvolutory Y, then Z SYS is skew-coninvolutory and E = ZE Z. Thus it suffices to consider a Jordan matrix J similar to J via a skew-coninvolutory matrix. We first consider the following technical lemma. Lemma 25. Suppose X =[x ij M n,k satisfies XJ k (α) = J X where J = β b b 2 b n 0 β b b b 0 0 β M n and b /= 0. (7) (a) If α/= β, then X = 0. (b) Suppose α = β. (i) If k n, then x ij = 0 whenever i + (k n)>j. (ii) If k<n,then x ij = 0 whenever i>j. That is, if k n, then X =[0P, where P M n is upper triangular; and if k<n,then X = [ P 0, where P M k is upper triangular. Proof. A computation reveals that (7) holds if and only if n x i,j + αx ij = βx ij + b m x i+m,j (8) m= for all i =,...,n and j =,...,k, where we adopt the convention that x pq = 0ifp = 0, q = 0orp>n. (a) Suppose α/= β. Write X =[x x 2 x k. Then XJ k (α) =[αx x + αx 2 x k + αx k and J X =[J x J x 2 J x k. Hence, J x = αx and (J αi)x = 0. Since α is not an eigenvalue of J, then J αi is nonsingular, hence, x must be zero. Now, J x 2 = x + αx 2 = αx 2, so that x 2 is also zero. Repeating this process yields x i = 0 for i =,...,k, and thus, X = 0. (b) Suppose α = β. Then (8) is equivalent to n x i,j = b m x i+m,j (9) m= for all i =,...,n, and j =,...,k. Examining (9) for i = n shows that x n,j = 0 for all j =,...,k, that is, x nj = 0 for j =,...,k. This will imply that x n,j = b x nj = 0 for

13 552 Ma.N.M. Abara et al. / Linear Algebra and its Applications 426 (2007) j =,...,k and hence x n,j = 0 for j =,...,k 2. Repeating this for i = n 2upto i = k implies that all the entries below x n r,k r, where r =,...,k, are zero. Hence, if k = n, then X is upper triangular; if k>n, then X =[0X, where X M n is upper triangular; and if k<n, then X = [ X2 X 3, where X 3 M k is upper triangular. Therefore, x ij = 0ifi + k n>j for all positive integers k and n. This proves (i). To prove (ii), notice that since x ij = 0 whenever i>j+ (n k), then (9) is equivalent to x i,j = n k+j m= b m x i+m,j for all i =,...,nand j =,...,k.when j =, (0) becomes 0 = n k+ m= b m x i+m,. Since x i = 0 whenever i>(n k) +, then () becomes b x n k+, = 0 when i = n k. Hence x n k+, = 0. Hence, if all the entries below x i+, are zero for a particular i, then () becomes b x i+, = 0 thus x i+, = 0. Therefore x i = 0 for i = 2,...,n.Suppose that for all q =,...,j, x iq = 0 whenever i>q.then (0) implies that n k+j m= b m x i+m,j = 0. If i = n k + j, then the sum (2) is just b x n k+j,j = 0, hence x n k+j,j = 0. Taking the sum (2) starting from row n k + j uptorowi = n k + j (n k) = j will yield b x i+,j = 0, hence x i+,j = 0. Therefore x i+,j = 0 for i = j,j +,...,n k + j, that is, x ij = 0 whenever i>j. For μ/= 0, μ b b 2 b n 0 μ. b... J n (μ) =.... b2,.... b 0 0 μ where b m = ( ) m (μ) (m+). Hence, if μ/= 0 and A =[a ij M n,k such that AJ k () = J n (μ) A, then we get a special case of Eq. (7). Thus we have the following assertion. Lemma 26. Let J M n such that J = m i= J i, where J i M ni, { j J kij ( i ), if i =, J i = [ ( ) j J kij ( i ) J kij, if i /=, i the i s are distinct and i j /= for i/= j. If X M n is such that XJ = J X, then X = m i= X i, where X i M ni. (0) () (2)

14 Ma.N.M. Abara et al. / Linear Algebra and its Applications 426 (2007) Proof. Partition X = (X ij ) conformal to J. The equality XJ = J X implies that X ij J j = J i Xij. Since the s are distinct and since i j /= whenever i/= j, then by Lemma 25, X ij = 0 whenever i/= j. Hence X = m i= X i, where X i M ni. Suppose E M n is similar to E and let e iθ,...,e iθ s, s+,,..., t,, with s+ i /= t ( ) be the distinct eigenvalues of E. Note that for i /=, if J k () occurs in J then so does J k. Hence, the Jordan canonical form of E may be written as J = s j= J e iθ j t j=s+ J j, [ ( where J iθ e j = j J kj (e iθ j ) and J j = j J kj ( j ) J kj ). IfX M n is such that XJ = j J X and X is skew-coninvolutory, then by Lemma 26, X = t j= X j and X j must be skewconinvolutory for all j. We consider what happens when there is an unpaired Jordan block J k (), where is on the unit circle. Lemma 27. Let = e iθ for some θ R and let J = r i= J k i () M n be such that there is an unpaired block J ki () for some i. Then J is not similar to J via a skew-coninvolutory matrix. Proof. Suppose that for some skew-coninvolutory X =[x ij M n we have XJ = J X and assume that there is an unpaired block of order k. We consider the following possibilities. (i) All the blocks are of the same size, that is, J = m J k () M mk, where m is necessarily odd since we are assuming that there is an unpaired block J k (). Partition X = (X ij ) conformal to J. Lemma 25 implies that X ij is a k-by-k upper triangular matrix for all i, j =,...,m.let P M mk be the permutation matrix obtained by interchanging the (mk i)th column with the (m i)kth column of I mk, i =, 2,...,m. Then [ P T B B XP = 2, where B 0 B 22 M m. (3) 22 Since X is skew-coninvolutory, then so is P T XP by Proposition 3. Hence B 22 is skew-coninvolutory. But B 22 is of odd dimension and thus cannot be skew-coninvolutory. (ii) There are Jordan blocks with size different from k. There are three cases: all the other Jordan blocks are of order less than k; all the other Jordan blocks are of order greater than k; and there are Jordan blocks of order less than k and greater than k. We prove only the third case since the proofs of the other two cases use similar arguments. Suppose J = J J 2 J 3, where J = ni <kj ni () M r, J 2 = lj >kj lj () M s and J 3 = m J k () M mk, where m is necessarily odd. Partition X = (X ij ) conformal to J. By Lemma 25, X 33 consists of k-by-k upper triangular blocks; X 3 consists of n i -by-k blocks, all of the form [0 V, where V M ni is upper triangular; [ W 0 [ Y 0 X 23 consists of l j -by-k blocks all of the form, where W M k is upper triangular; X 3 consists of k-by-n i blocks, all of the form, where Y M ni is upper triangular; and X 32 consists of k-by-l j blocks, all of the form [0 Z, where Z M k is upper triangular.

15 554 Ma.N.M. Abara et al. / Linear Algebra and its Applications 426 (2007) Let Q = I r I s P, where P is the permutation matrix described above. Then X X 2 X 3 P Q T XQ = X 2 X 22 X 23 P, (4) P T X 3 P T X 32 P T X 33 P such that the last m rows of P T X 3 are zero; the last m rows of P T X 32 are zero except for every (l j )th column; every (l j )th row of X 23 P is zero; and P T X 33 P is of the form described [ in Eq. (3). Hence, Q T XQ may be written as C C 2 C 2 C 22, where C 22 = B 22 M m, C 2 = [ 0 m,r D U 0,m 0 u 0 u t 0 m,m(k ) and C 2 =.., U t 0,m B 2 such that u i C m, D M r,m and U i M li,m. If X is skew-coninvolutory, then by Proposition 3,soisQ T XQ, thus C 2 C 2 + C 22 C 22 = I m. Since C 2 C 2 = 0, then C 22 must be skew-coninvolutory. But this is a contradiction since C 22 is of odd dimension, hence, X cannot be skew-coninvolutory. Now, we exclude all matrices with unpaired J k (e iθ ). Lemma 28. Let J be as in Lemma 26 and suppose that there exists an unpaired block corresponding to = e iθ with θ R. Then J is not similar to J via a skew-coninvolutory matrix. Proof. Let J be as in Lemma 26 and such that there is an unpaired block corresponding to. If AJ = J A, then A = A A r and notice that A is skew-coninvolutory if and only if each A i is skew-coninvolutory. From Lemma 27, A is not skew-coninvolutory. Thus J is not similar to J via a skew-coninvolutory matrix. By Theorem 24 and Lemmas 25 28, we have the following theorem. Theorem 29. A M n is ( similar to A ( via )) a skew-coninvolutory matrix if and only the Jordan canonical form of A is J k () J k A similar to A Suppose A M n is similar to A. The following theorem shows that the matrix of similarity may be chosen to be coninvolutory. Hence, if a nonsingular matrix A is similar to A, we prove in the following theorem that A is a ψ S -orthogonal matrix for some S C n.

16 Ma.N.M. Abara et al. / Linear Algebra and its Applications 426 (2007) Theorem 30. Let A M n be given. The following are equivalent: (a) A is similar to A. (b) A is similar to a real matrix R M n. (c) A is similar to a real R via a coninvolutory matrix. (d) A is similar to A via a coninvolutory matrix. Proof. Since any matrix A is similar to its transpose, if A is similar to A, then A is similar to A. By Theorem 4..7 of [, A is similar to a real matrix, hence (a) implies (b). Suppose A = X RX for some real R and a nonsingular X. By the real-coninvolutory decomposition, there exists a real S and a coninvolutory E such that X = SE. Then A = E S RSE and S RS is real, hence (c) follows. Suppose A = E RE for a real R and a coninvolutory E. Then A = E RE = ERE, and thus R = E AE. This implies A = (E 2 ) AE 2, that is, A is similar to A via a coninvolutory matrix and thus (c) implies (d). One checks that (d) implies (a). Suppose A M n is similar to A. Then whenever J k () is in the Jordan canonical form of A, so is J k (). Notice that J J k () J k () is similar to J via the skew-coninvolutory matrix J 2k. Using arguments similar to Theorem 24, we obtain the following class of matrices which satisfies Theorem 30. Theorem 3. Let A M n have Jordan canonical form (J k () J k ()). Then A is similar to A via a skew-coninvolutory matrix. If A is similar to A and is a real eigenvalue of A, then the Jordan blocks corresponding to need not come in pairs. Thus begs the question whether A is similar to A via a skew-coninvolutory matrix if there is an unpaired Jordan block J k () for some R. Observe that if A M n,k and μ C such that AJ k () = J n (μ)a, then A would be of the form given in Eq. (7). Hence we have the following analogous results for the case when A is similar to A. Lemma 32. Let J M n such that J = m r= J r, where J r M nr, { j J krj ( r ) if r R, J r = j [J krj ( r ) J krj ( r ) if r / R, the r s are distinct and r /= s if r /= s. If X M n such that XJ = JX, then X = m r= X r, where X r M nr. Lemma 33. Let R and let J = r i= J k i () M n such that there is an unpaired block J ki () for some i. Then J is not similar to J via a skew-coninvolutory matrix. Theorem 34. A M n is similar to A via a skew-coninvolutory matrix if and only the Jordan canonical form of A is J = (J k () J k ()). Thus A is a ψ S -orthogonal matrix for some S D n if and only if its Jordan canonical form consists of pairs of J k () J k ().

17 556 Ma.N.M. Abara et al. / Linear Algebra and its Applications 426 (2007) A similar to A We consider a related result for the case when A M n is similar to A. Theorem 35. Let A M n. The following are equivalent: (a) A is similar to A. (b) A = X PX,where P is pure imaginary. (c) A = E PE,where E is coninvolutory and P is pure imaginary. (d) A = E ( A)E, where E is coninvolutory. Proof. If A is similar to A, then whenever J k () occurs in the Jordan canonical form of A, so does J k ( ). If = ia for some real a, then =. Hence, the Jordan blocks in the Jordan canonical form of A are J k (ia) for some a R or J k () J k ( ), whenever is not pure imaginary and J k (ia) is similar to the pure imaginary matrix ij k (a). On the other hand, since [ iik I k I k ii k [ Jk () 0 0 J k () [ [ iik I k Jk () J = k () i(j k () + J k ()) I k ii k i(j k () + J k ()) J k () J k () which is pure imaginary, then (J k () J k ( )) is similar to a pure imaginary matrix. Hence A = X PX, for some nonsingular X and a pure imaginary P. Suppose A = X PX, where P is pure imaginary. Let X = RE be a real-coninvolutory decomposition of X. Then A = E (R PR)E, and R PR is still pure imaginary. Suppose A=E PE, where E is coninvolutory and P is pure imaginary. Then A=E P E = EPE which implies that P = E ( A)E. Thus, A = E PE = (E 2 ) ( A)E 2. Therefore, A is similar to A via a coninvolutory matrix. One checks that (d) implies (a). Thus, if A is a nonsingular matrix similar to A, then, by Theorem 35, ψ S (A) = A for some S C n. We present a class of matrices satisfying Theorem 35. Theorem 36. Let A M n have Jordan canonical form (J k () J k ( )). Then A is similar to A via a skew-coninvolutory matrix. Proof. Observe that J k () J k () = J 2k [ (J k () J k ()) J 2k. Using similar arguments in the proof of Theorem 24, we conclude that A is similar to A via a skew-coninvolutory matrix. If A is similar to A and Ri is an eigenvalue of A, then the Jordan blocks corresponding to need not come in pairs. Suppose X M n,k such that XJ k () = J n (μ)x. Then X will be of the form given in Lemma 25. We follow the arguments in the case when A is similar to A to prove the converse of Theorem 36. Lemma 37. Let J M n such that J = m r= J r, where J r M nr, { j J krj ( r ) if r ir, J r = j [J krj ( r ) J krj ( r ) if r / ir,,

18 Ma.N.M. Abara et al. / Linear Algebra and its Applications 426 (2007) the r s are distinct and r /= s if r /= s. If X M n such that XJ = JX,then X = m r= X r, where X r M nr. Lemma 38. Let C \ R and let J = r i= J k i () M n such that there is an unpaired block J ki () for some i. Then J is not similar to J via a skew-coninvolutory matrix. Theorem 39. A M n is similar to A via a skew-coninvolutory matrix if and only the Jordan canonical form of A is J = (J k () J k ( )). Theorem 39 implies that a nonsingular matrix A satisfies ψ S (A) = A for some S D n if and only if the Jordan canonical form of A consists of pairs of J k () J k ( ). References [ R.A. Horn, C.R. Johnson, Matrix Analysis, Cambridge University Press, New York, 985. [2 R.A. Horn, C.R. Johnson, Topics in Matrix Analysis, Cambridge University Press, New York, 985. [3 R.A. Horn, D.I. Merino, A real-coninvolutory analog of the polar decomposition, Linear Algebra Appl. 90 (993) [4 R.A. Horn, D.I. Merino, Contragredient equivalence: a canonical form and some applications, Linear Algebra Appl. 24 (995)