On a problem of Halmos: Unitary equivalence of a matrix to its transpose

Transcription

1 On a problem of Halmos: Unitary equivalence of a matrix to its transpose Stephan Ramon Garcia Pomona College Claremont, California March 18, 2011 Abstract Halmos asked whether every square complex matrix is unitarily equivalent to its transpose (UET). Ad hoc examples indicate that the answer is no. In this talk, we give a complete characterization of matrices which are UET. Surprisingly, the naïve conjecture that a matrix is UET if and only if it is unitarily equivalent to a complex symmetric (i.e., self-transpose) matrix is true in dimensions n 7 but false for n 8. In particular, unexpected building blocks begin to appear in dimensions 6 and 8. This is joint work with James E. Tener (UC Berkeley). Partially supported by NSF Grant DMS (SRG) and a NSF Graduate Research Fellowship (JET).

2 A Halmos Problem Problem (Halmos, LAPB - Problem 159) Is every square matrix unitarily equivalent to its transpose (UET)?

3 A Halmos Problem Problem (Halmos, LAPB - Problem 159) Is every square matrix unitarily equivalent to its transpose (UET)? Solution (Halmos) Ad-hoc methods show that is not UET.

4 A Halmos Problem Problem (Halmos, LAPB - Problem 159) Is every square matrix unitarily equivalent to its transpose (UET)? Solution (Halmos) Ad-hoc methods show that is not UET. Can we characterize those matrices which are UET?

5 Who Cares? Why is this interesting?

6 Who Cares? Why is this interesting? It s technically a Halmos Problem

7 Who Cares? Why is this interesting? It s technically a Halmos Problem Every square matrix is similar to its transpose, making the problem somewhat difficult.

8 Who Cares? Why is this interesting? It s technically a Halmos Problem Every square matrix is similar to its transpose, making the problem somewhat difficult. Linear preservers of the numerical range are of the form X UXU or X UX t U (where U is unitary). Thus we characterize the fixed points of linear preservers in the nontrivial case.

9 Who Cares? Why is this interesting? It s technically a Halmos Problem Every square matrix is similar to its transpose, making the problem somewhat difficult. Linear preservers of the numerical range are of the form X UXU or X UX t U (where U is unitary). Thus we characterize the fixed points of linear preservers in the nontrivial case. It has an obvious answer, which is totally wrong.

10 Who Cares? Why is this interesting? It s technically a Halmos Problem Every square matrix is similar to its transpose, making the problem somewhat difficult. Linear preservers of the numerical range are of the form X UXU or X UX t U (where U is unitary). Thus we characterize the fixed points of linear preservers in the nontrivial case. It has an obvious answer, which is totally wrong. Funny things happen once you hit dimensions 6 and 8.

11 A näive conjecture Definition If T is unitarily equivalent to a complex symmetric (i.e., selftranspose) matrix, then T is UECSM.

12 A näive conjecture Definition If T is unitarily equivalent to a complex symmetric (i.e., selftranspose) matrix, then T is UECSM. A Coincidence? Halmos matrix is the simplest example of a matrix which is not UECSM.

13 A näive conjecture Definition If T is unitarily equivalent to a complex symmetric (i.e., selftranspose) matrix, then T is UECSM. A Coincidence? Halmos matrix is the simplest example of a matrix which is not UECSM. Näive Conjecture UET UECSM ( trivial)

14 UECSMs are Tricky Exactly one of the following is UECSM: ( 0 7 ) ( ) ( ) ( ) ( )

15 UECSMs are Tricky Exactly one of the following is UECSM: ( 0 7 ) ( ) ( ) ( ) ( ) Which one is UECSM?

16 UECSMs are Tricky Exactly one of the following is UECSM: ( 0 7 ) ( ) ( ) ( ) ( ) Which one is UECSM? The fourth matrix is unitarily equivalent to i i i i

17 A Gallery of UECSMs = = = = i ( 1 7 ) 74 2 ( 3i 2 74 ) ( 1 2 i ) i 1 2 (11 ) 82 ( 82 5 ) 82 2i ( ( 1 5 ) i i i ) ( 3i 2 74 ) ( 2i ) i ( ) ( ) 187 ( 1 2 i ) ( ) 74 ( 1 2 i ) ( ) 82 i ( ) i i

18 Another Contender?

19 Skew-Hamiltonian Matrices Definition If B t = B and D t = D, then ( ) A B T = D A t is a skew-hamiltonian matrix (SHM).

20 Skew-Hamiltonian Matrices Definition If B t = B and D t = D, then ( ) A B T = D A t is a skew-hamiltonian matrix (SHM). If T is unitarily equivalent to a SHM, then T is UESHM.

21 Skew-Hamiltonian Matrices Definition If B t = B and D t = D, then ( ) A B T = D A t is a skew-hamiltonian matrix (SHM). If T is unitarily equivalent to a SHM, then T is UESHM. UESHM UET If T is skew-hamiltonian, then T = JT t J where ( ) 0 I J =. I 0

22 Reducibility of Skew-Hamiltonian Matrices Theorem (Waterhouse, 2004) If T is skew-hamiltonian, then T is similar to A A t for some A.

23 Reducibility of Skew-Hamiltonian Matrices Theorem (Waterhouse, 2004) If T is skew-hamiltonian, then T is similar to A A t for some A. Proposition If T is skew-hamiltonian and 6 6 or smaller, then T is reducible.

24 Reducibility of Skew-Hamiltonian Matrices Theorem (Waterhouse, 2004) If T is skew-hamiltonian, then T is similar to A A t for some A. Proposition If T is skew-hamiltonian and 6 6 or smaller, then T is reducible. Pf. Sketch, 6 6 case. QT = TQ and Q = Q lead to a real linear system, which is too large to consider symbolically.

25 Reducibility of Skew-Hamiltonian Matrices Theorem (Waterhouse, 2004) If T is skew-hamiltonian, then T is similar to A A t for some A. Proposition If T is skew-hamiltonian and 6 6 or smaller, then T is reducible. Pf. Sketch, 6 6 case. QT = TQ and Q = Q lead to a real linear system, which is too large to consider symbolically. Exploiting symmetries and using various reductions, one obtains an equivalent real linear system.

26 Reducibility of Skew-Hamiltonian Matrices Theorem (Waterhouse, 2004) If T is skew-hamiltonian, then T is similar to A A t for some A. Proposition If T is skew-hamiltonian and 6 6 or smaller, then T is reducible. Pf. Sketch, 6 6 case. QT = TQ and Q = Q lead to a real linear system, which is too large to consider symbolically. Exploiting symmetries and using various reductions, one obtains an equivalent real linear system. Mathematica s symbolic MatrixRank command shows that the rank is 20, whence a nonscalar Q exists.

27 Irreducible skew-hamiltonians exist if n 8 Numerical Evidence Computation indicates that generic 8 8 SHMs are irreducible. Finding a scalable family of provably irreducible SHMs is difficult.

28 Irreducible skew-hamiltonians exist if n 8 Numerical Evidence Computation indicates that generic 8 8 SHMs are irreducible. Finding a scalable family of provably irreducible SHMs is difficult. Proposition For d 4, the matrix T = 1 A = 2... d ( ) A B, where 0 A, B = is skew-hamiltonian and irreducible ,

29 Putting It All Together

30 An Instructive Example A Potential Pitfall T = is UET, UECSM, and UESHM.

31 An Instructive Example A Potential Pitfall T = is UET, UECSM, and UESHM. However, A = none of these properties. ( ) itself has 0 0 0

32 An Instructive Example A Potential Pitfall T = is UET, UECSM, and UESHM. However, A = none of these properties. Remarks ( ) itself has If T = A A t, then T is UECSM, UESHM, and UET.

33 An Instructive Example A Potential Pitfall T = is UET, UECSM, and UESHM. However, A = none of these properties. Remarks ( ) itself has If T = A A t, then T is UECSM, UESHM, and UET. Neither class UECSM nor UESHM is closed under restriction to direct summands.

34 An Instructive Example A Potential Pitfall T = is UET, UECSM, and UESHM. However, A = none of these properties. Remarks ( ) itself has If T = A A t, then T is UECSM, UESHM, and UET. Neither class UECSM nor UESHM is closed under restriction to direct summands. This issue is important only in dimensions 6

35 Final Answer Theorem (SRG, JET) T is UET if and only if T is unitarily equivalent to a direct sum of:

36 Final Answer Theorem (SRG, JET) T is UET if and only if T is unitarily equivalent to a direct sum of: 1 irreducible CSMs

37 Final Answer Theorem (SRG, JET) T is UET if and only if T is unitarily equivalent to a direct sum of: 1 irreducible CSMs 2 irreducible SHMs (automatically 8 8 or larger),

38 Final Answer Theorem (SRG, JET) T is UET if and only if T is unitarily equivalent to a direct sum of: 1 irreducible CSMs 2 irreducible SHMs (automatically 8 8 or larger), 3 matrices of the form A A t where A is irreducible and neither UECSM nor UESHM (automatically 6 6 or larger).

39 Final Answer Theorem (SRG, JET) T is UET if and only if T is unitarily equivalent to a direct sum of: 1 irreducible CSMs 2 irreducible SHMs (automatically 8 8 or larger), 3 matrices of the form A A t where A is irreducible and neither UECSM nor UESHM (automatically 6 6 or larger).

40 Final Answer Theorem (SRG, JET) T is UET if and only if T is unitarily equivalent to a direct sum of: 1 irreducible CSMs 2 irreducible SHMs (automatically 8 8 or larger), 3 matrices of the form A A t where A is irreducible and neither UECSM nor UESHM (automatically 6 6 or larger). Moreover, the unitary orbits of these classes are pairwise disjoint.

41 Final Answer Theorem (SRG, JET) T is UET if and only if T is unitarily equivalent to a direct sum of: 1 irreducible CSMs 2 irreducible SHMs (automatically 8 8 or larger), 3 matrices of the form A A t where A is irreducible and neither UECSM nor UESHM (automatically 6 6 or larger). Moreover, the unitary orbits of these classes are pairwise disjoint. Corollary If T M n (C) is UET and n 7, then T is UECSM.

42 Final Answer Theorem (SRG, JET) T is UET if and only if T is unitarily equivalent to a direct sum of: 1 irreducible CSMs 2 irreducible SHMs (automatically 8 8 or larger), 3 matrices of the form A A t where A is irreducible and neither UECSM nor UESHM (automatically 6 6 or larger). Moreover, the unitary orbits of these classes are pairwise disjoint. Corollary If T M n (C) is UET and n 7, then T is UECSM. If n 5, then T is unitarily equivalent to a direct sum of irreducible CSMs.

43 A Trace Criterion Theorem (Pearcy, Sibirskĭı, 1968) For 3 3 matrices, A = B if and only if the traces of X, X 2, X 3, X X, X X 2, X 2 X 2, X X 2 X 2 X agree for X = A and X = B.

44 A Trace Criterion Theorem (Pearcy, Sibirskĭı, 1968) For 3 3 matrices, A = B if and only if the traces of X, X 2, X 3, X X, X X 2, X 2 X 2, X X 2 X 2 X agree for X = A and X = B. Proposition (SRG, JET) If T is 3 3, then T is UECSM if and only if tr[t T (T T TT )TT ] = 0.

45 A Trace Criterion Theorem (Pearcy, Sibirskĭı, 1968) For 3 3 matrices, A = B if and only if the traces of X, X 2, X 3, X X, X X 2, X 2 X 2, X X 2 X 2 X agree for X = A and X = B. Proposition (SRG, JET) If T is 3 3, then T is UECSM if and only if tr[t T (T T TT )TT ] = 0. Proof. Check if T = T t. Only the final word is not satisfied trivially.

46 Bibliography 1 Balayan, L., Garcia, S.R., Unitary equivalence to a complex symmetric matrix: geometric criteria, Oper. Matrices 4 (2010), No. 1, Garcia, S.R., Poore, D.E., Wyse, M.K., Unitary equivalent to a complex symmetric matrix: a modulus criterion, Oper. Matrices (to appear). 3 Garcia, S.R., Tener, J.E., Unitary equivalence of a matrix to its transpose, J. Operator Theory (to appear). 4 Garcia, S.R., Wogen, W., Complex symmetric partial isometries, J. Funct. Analysis 257 (2009), Garcia, S.R., Wogen, W., Some new classes of complex symmetric operators, Trans. Amer. Math. Soc. 362 (2010), Halmos, P.R., Linear algebra problem book, The Dolciani Mathematical Expositions, 16. Mathematical Association of America, Washington, DC, Pearcy, C., A complete set of unitary invariants for 3 3 complex matrices, Trans. Amer. Math. Soc. 104 (1962), Sibirskiĭ, K.S., A minimal polynomial basis of unitary invariants of a square matrix of order three (Russian), Mat. Zametki 3 (1968), Tener, J.E., Unitary equivalence to a complex symmetric matrix: an algorithm, J. Math. Anal. Appl. 341 (2008), Waterhouse, William C., The structure of alternating-hamiltonian matrices, Linear Algebra Appl. 396 (2005),