Matrices That Commute with Their Conjugate and Transpose

Home , Conjugate transpose

Advances in Linear Algebra & Matrix Theory, 2013, 3, 22-25 http://dx.doi.org/10.4236/alamt.2013.33005 Published Online September 2013 (http://www.scirp.org/journal/alamt)

Geoffrey Goodson Towson University, Towson, USA Email: [email protected]

Received June 25, 2013; revised July 23, 2013; accepted August 6, 2013

Copyright © 2013 Geoffrey Goodson. 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.

ABSTRACT

** TT It is known that if A M n is normal AAAA , then AAAA if and only if AAAA . This leads to the question: do both AAAA and AATT AA imply that A is normal? We give an example to show that this is false when n  4 , but we show that it is true when n  2 and n  3 .

Keywords: Normal Matrix; Matrix Commuting with Its Conjugate and Transpose

Introduction and Results 2011). My solution for n  2 appeared in the spring 2012 issue, but no solution for the case n  3 has ever Let A be an n -by- n normal matrix, i.e., A is a been given. In this paper, we give the solution for the complex square matrix A M , with the property that n  case n  3 , and for completeness, we also give the solu- AA* AA*, where A* AT is the conjugate-transpose tion for n  2 . Specifically we prove: of A . The Fuglede-Putnam Theorem tells us that if ** Theorem 1 If A M n , n  2 or n  3, then AB = BA for some B M, then A BBA . Suppose TT n AAAA and AA AA imply that A is normal. that AA AA, where A is the conjugate of the matrix Proof. We need the following preliminary result, A (so we take the complex conjugate of every entry of which is a direct consequence of Theorem 2.3.6 in [3] A ). Then taking the transpose gives (using the fact that for A M n , A BiC where B ATTAAAAAAAAAAA TT *T T* T T, and C are real then AAAA if and only if BC CB ), and stated explicitly in [1,2]. from the the Fuglede-Putnam Theorem. In a similar way, Theorem 2 Let A M , n  3 , with AAAA . we see that if AAAATT , then AAAA , so these two n Then there exists a real orthogonal matrix QM n  statements are equivalent when A is normal. The ques- T such that QAQ is of the form: tion arose in [2], whether the conditions A    AAAAand AAAATT 1  0 A      2  imply the third condition AAAA** , so that A is nor-  00A3   , mal.    This is false when n  4 . In fact, any matrix of the     00 0Ak IIab cd ab   form A  , where Iab  , 0 Iab ba where each Ai , 1 ik (for some k ) is a 1-by-1 abcd,,,  , cd220 , c and d not both zero, matrix or a 2-by-2 matrix. T has the property that both AA AA and ATTAAA , Example 1. Note that if A QQ, Q real ortho- TT but A is not normal. In this paper, we prove that if gonal, AAAA and AA AA if and only if TT T A M where n  2 or n  3 , then these conditions  and . Also note that if A  A n * do imply that A is normal. This result was first pro- and AAAA . then A is normal since A  A in this posed as a problem by the current author in the Interna- case. TT tional Linear Algebra Society journal IMAGE (fall Lemma 1 If A M 2 with AAAA , then A is

ab or either symmetric or of the form ,,ab . ba abx222 xy x   ab xy a22 b y 2 y Proof. Suppose that Aabcd,,,,  , with     2  cd  xy TT AAAA , then ab220 axby   a2b2 a c bd a22 c ab cd  0.ab22 bxay    2222  22 2 ac bd c d ab dc b d  ax by ay bx x y   Hence bc22 and ab cd ac bd . It follows that xy  0 , and A is normal. Case 1. bc , so that A is symmetric. axy Case 2. bc, then ab bd ab bd or Case 2: If 0 bz then ab  bd . If b  0 , then A is symmetric. If b  0 ,   00c ab   ad and A  . axya00   a 00  axy  ba 0000,bzxb   xb  bz  Propositi on 1 If A M with AAAATT and       2 00cyzc   yzc  00 c  AAAA , then A is norm al.       Proof. From the Lemma 1, we have two cases. If or 222 ab axybxyzcy   A  , ab,  , then A is normal. On the  22  ba  bx yz b z cz  2  other hand, if A is symmetric with AAAA , then  cy cz c  since A*  A in this case, we must have AA** AA, aaxay2  so A is norm al.   ax x22 b xy  bz . Ex ample 2. We now look at the case of A M 3 . We   start with a lemma:  222 ay xy bz c y z  Lemma 2 Suppose A M with AA AA , 22 22 22 3 Hence xy  0 , x  z , yz0 , and AAAATT and A QQT for some real o rtho gonal x  iy , x  z , y  iz and also ay cy , so y  0 (giving  diagonal and A normal) or ac . Sup- matrix Q M3  where  is of one of the two pose y  0 so that ac . forms: see Equation (1). Case 2(a). If xz 0 , then then A is normal.   ax bx yz y b a and x ib  a  z so that abx  aibaba    Proof. Case 1:  ba y. Now we require    0.biab   However, this matrix 00     00 a  TT       , so that also has the property that    , which gives in abxa b00a b abxEquation (2). It follows from equating the entries in the  ba y b a00 b a ba y, (1, 2) position 22 22 2 00 xy xy 00a b22 ab  a b ab , or ab0 ,

abx axy   ba y,or  0 bz , ab ,,,,,,cxyz  , (1) 00 00c

a222 i a b2 ab 2 ab  23 ab a222 b a i a22 b2 ab 2 ab  23 ab a2 b 2      2222 22  02biabab 0 biaba2b (2)   00 aa2 0 0 2   

so ab , and hence  is diagonal and A is normal. ay y d e ye ay y d e ye Case 2(b). This is where xz  0 , and since ax bx xy we have yab, so that and simplifying gives yd  a ya  d, so if ad a iab ab  yad  we have  0,biab  and this is treated in a yad Equating entries in the (2, 3) position gives: 00 a  similar way. y ea dyyeyea   dyye, TT Proposition 2 If A M 3 with AA AA and AAAA , then A is normal. and this reduces to: yd  a  yd a, so if ad , Proof. We show that every case reduces to the case of yad  , contradicting the above. We conclude that the Lemma 2. From Theorem 1, every matrix  with yad T A QQ (Q real orthogonal) can be chosen t o be ad and  is of the form one of the following three forms:  aiaeiy  axy abx      ia e a y , (I) 0 bz, (II) cd y, or (III)  00e  00c 00e and we can apply Lemma 2. The other possibility is that axy  y  0  x , so that bc  and  is either of the form  0 bc. ab0 0 de   bd0 , a symmetric matrix (when cb ), or of We have dealt with Case (I) in Lemma 2, so consider 00e Case (II): TT    gives in Equation (3).  abiy It follows that bxc222, cyb222, and the form  ba y (when cb , since in this xy220 , so that x iy .    00e  Case 1. xi0y, then since bx dy ey ,   biy dy ey , so bide. case ad ). Case 2. xiy 0 , then bx dy ey gives Also xeaxcy gives xeaxcix , so that ciea=  , so  has the form bied    , and ax cy xe gives ciae  , so aieiyd  that  has the form   iea d y  aiediy     00e  ia  e d y.   Now we use the fa ct that    . This gives in E qua-  00e  tion (4). On equat ing entries in the (1, 3) position we have: We proceed exact ly as in Ca se 1 to re duce  to the

abxacbxyxeac222 d 2 2 abcdaxcy    222 22  ac bd xy c  d y ey  ab  cd b  d bx  dy (3) 2222 xe ye e ax cy xb yd x y e 

adeeaiadeidd2    eiayiydeiye     iaea idea  ead   e d2 yea  dyye    2 00e  a2  d e e  a ia d  e id d  e iay  iy d  e iye    ia e  a id e  a e  a d  e d2 y e  a dy ye (4)   00e 2 

situation of Lemma 2. doi:10.3103/S0278641910020019 axy [2] G. R. Goodson and R. A. Horn, “Canonical Forms for Normal Matrices That Commute with Their Complex In Case (III), where 0 bc, we proceed ex-  Conjugate,” Linear Algebra and Its Applications, Vol. 0 de 430, No. 4, 2009, pp. 1025-1038. actly as in Case (II) to deduce the result. doi:10.1016/j.laa.2008.09.039 [3] R. A. Horn and C. R. Johnson, “Matrix Analysis,” Cam- bridge University Press, New York, 1985. REFERENCES doi:10.1017/CBO9780511810817

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