Dennis I. Merino

Contragredient equivalence: A canonical form and some applications

Abstract Let A and C be m -by- n complex matrices, and let B and D be n -by- m complex matrices. The pair ( A , B ) is contragrediently equivalent to the pair ( C , D ) if there are square nonsingular complex matrices X and Y such that XAY −1 = C and YBX −1 = D . Contragredient equivalence is a common generalization of four basic equivalence relations: similarity, consimilarity, complex orthogonal equivalence, and unitary equivalence. We develop a complete set of invariants and an explicit canonical form for contragredient equivalence and show that ( A , A T ) is contragrediently equivalent to ( C , C T ) if and only if there are complex orthogonal matrices P and Q such that C = PAQ . Using this result, we show that the following are equivalent for a given n -by- n complex matrix A : 1. (1) A = QS for some complex orthogonal Q and some complex symmetric S ; 2. (2) A T A is similar to AA T ; 3. (3) ( A , A T ) is contragrediently equivalent to ( A T , A ); 4. (4) A = Q 1 A T Q 2 for some complex orthogonal Q 1 , Q 2 ; 5. (5) A = PA T P for some complex orthogonal P . We then consider a linear operator φ on n -by- n complex matrices that shares the following properties with transpose operators: for every pair of n -by- n complex matrices A and B , (a) φ. preserves the spectrum of A , (b) φ ( φ ( A )) = A , and (c) φ ( AB ) = φ ( B ) φ ( A ). We show that ( A , φ ( A )) is contragrediently similar to ( B , φ ( B )) if and only if A = X 1 BX 2 for some nonsingular X 1 , X 2 that satisfy X −1 1 = φ ( X 1 ) and X −1 2 = φ ( X 2 ). We also consider a factorization of the form A = XY , where X −1 = φ ( X ) and Y = φ ( Y ). We use the canonical form for the contragredient equivalence relation to give a new proof of a theorem of Flanders concerning the relative sizes of the nilpotent Jordan blocks of AB and BA . We present a sufficient condition for the existence of square roots of AB and BA and close with a canonical form for complex orthogonal equivalence.

The Jordan Canonical Forms of complex orthogonal and skew-symmetric matrices

Abstract We study the Jordan Canonical Forms of complex orthogonal and skew-symmetric matrices, and consider some related results.

A real-coninvolutory analog of the polar decomposition

Abstract We study properties of coninvolutory matrices (EĒ = I), and derive a canonical form under similarity as well as a canonical form under unitary consimilarity for them. We show that any complex matrix has a coninvolutory dilation, and we characterize the minimum size of a coninvolutory dilation of a square matrix. We characterize the m-by-n complex matrices A that can be factored as A = RE with R real and E coninvolutory, and we discuss the uniqueness of this factorization when A is square and nonsingular.

Littlewood's algorithm and quaternion matrices

Abstract A strengthened form of Schurs triangularization theorem is given for quaternion matrices with real spectrum (for complex matrices it was given by Littlewood). It is used to classify projectors ( A 2 = A ) and self-annihilating operators ( A 2 =0 ) on a quaternion unitary space and examples of unitarily wild systems of operators on such a space are presented. Littlewoods algorithm for reducing a complex matrix to a canonical form under unitary similarity is extended to quaternion matrices whose eigenvalues have geometric multiplicity 1.

Quasi-real normal matrices and eigenvalue pairings

Abstract For square complex matrices A and B of the same size, commutativity-like relations such as AB=±BA, AB=±BA ∗ , AB=±BAT, AB=±BTA, etc., often cause a special structure of A to be reflected in some special structure for B. We study eigenvalue pairing theorems for B when A is quasi-real normal (QRN), a class of complex matrices that is a natural generalization of the real normal matrices. A new canonical form for QRN matrices is an essential tool in our development.

Linear operators preserving unitary t-congruence (orthogonal similarity) on complex (real) matrices

Two complex (real) square matrices A and B are said io be unitarily t-congruent (orthogonally similar) it there exists a unitary (an orthogonal) matrix U such that A=UBU 1 We characterize those linear operators that preserve unitary t-congruence on complex matrices and those linear operators that preserve orthogonal similarity on real matrices. This answers a question raised in a paper by Y. P. Hong, R. A. Horn and the first author.

Each n-by-n matrix with n > 1 is a sum of 5 coninvolutory matrices

Abstract An n × n complex matrix A is called coninvolutory if A ¯ A = I n and skew-coninvolutory if A ¯ A = − I n (which implies that n is even). We prove that each matrix of size n × n with n > 1 is a sum of 5 coninvolutory matrices and each matrix of size 2 m × 2 m is a sum of 5 skew-coninvolutory matrices. We also prove that each square complex matrix is a sum of a coninvolutory matrix and a condiagonalizable matrix. A matrix M is called condiagonalizable if M = S ¯ − 1 D S in which S is nonsingular and D is diagonal.

Distances between the graphs of matrices

The graph of a complex m-by-n matrix A is the range of [I AT]T in Cm+n. We find explicit formulae for the distance between the graphs of two m-by-n matrices with respect to two classes of metrics on the set of subspaces of Cm+n. The singular values of the contraction g(A, B)  (I+A∗ A)−12(I+A∗ B)(I+B∗ B)−12 play a key role. We study transformations that preserve various distances between graphs, and we give some inequalities between different distance functions.

Linear Operators Preserving Complex Orthogonal Equivalence on Matrices

Two complex matrices

Genetic Algorithms: Theory And Application

A