Canonical matrices of isometric operators on indefinite inner product spaces

We give canonical matrices of a pair (A,B) consisting of a nondegenerate form B and a linear operator A satisfying B(Ax,Ay)=B(x,y) on a vector space over F in the following cases:

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F is an algebraically closed field of characteristic different from 2 or a real closed field, and B is symmetric or skew-symmetric;

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F is an algebraically closed field of characteristic 0 or the skew field of quaternions over a real closed field, and B is Hermitian or skew-Hermitian with respect to any nonidentity involution on F.

These classification problems are wild if B may be degenerate. We use a method that admits to reduce the problem of classifying an arbitrary system of forms and linear mappings to the problem of classifying representations of some quiver. This method was described in [V.V. Sergeichuk, Classification problems for systems of forms and linear mappings, Math. USSR-Izv. 31 (3) (1988) 481-501].     

Simultaneous block diagonalization of two real symmetric matrices

The simultaneous diagonalization of two real symmetric (r.s.) matrices has long been of interest. This subject is generalized here to the following problem (this question was raised by Dr. Olga Taussky-Todd, my thesis advisor at the California Institute of Technology): What is the first simultaneous block diagonal structure of a nonsingular pair of r.s. matrices ? For example, given a nonsingular pair of r.s. matrices S and T, which simultaneous block diagonalizations X′SX = diag(A₁, , A_k), X′TX = diag(B₁,, B_k) with dim A_i = dim B_i and X nonsingular are possible for 1 ? k ? n; and how well defined is a simultaneous block diagonalization for which k, the number of blocks, is maximal? Here a pair of r.s. matrices S and T is called nonsingular if S is nonsingular.If the number of blocks k is maximal, then one can speak of the finest simultaneous block diagonalization of S and T, since then the sizes of the blocks A_i are uniquely determined (up to permutations) by any set of generators of the pencil P(S, T) = {aS + bT|a, tb ε R} via the real Jordan normal form of S^?1T. The proof uses the canonical pair form theorem for nonsingular pairs of r.s. matrices. The maximal number k and the block sizes dim A_i are also determined by the factorization over C of ? (λ, μ) = det(λS + μT) for λ, μ ε R.     

Independence for partition regular equations

A matrix A is said to be partition regular (PR) over a subset S of the positive integers if whenever S is finitely coloured, there exists a vector x, with all elements in the same colour class in S, which satisfies Ax=0. We also say that S is PR for A. Many of the classical theorems of Ramsey Theory, such as van der Waerden's theorem and Schur's theorem, may naturally be interpreted as statements about partition regularity. Those matrices which are partition regular over the positive integers were completely characterised by Rado in 1933.Given matrices A and B, we say that A Rado-dominates B if any set which is PR for A is also PR for B. One trivial way for this to happen is if every solution to Ax=0 actually contains a solution to By=0. Bergelson, Hindman and Leader conjectured that this is the only way in which one matrix can Rado-dominate another. In this paper, we prove this conjecture for the first interesting case, namely for 1×3 matrices. We also show that, surprisingly, the conjecture is not true in general.     

A criterion of diagonalizability of a pair of matrices over the ring of principal ideals by common row and separate column transformations

We establish that a pair A, B, of nonsingular matrices over a commutative domain R of principal ideals can be reduced to their canonical diagonal forms D ^A and D ^B by the common transformation of rows and separate transformations of columns. This means that there exist invertible matrices U, V _A, and V _B over R such that UAV _a=D^A and UAV _B=D^B if and only if the matrices B _*A and D _* ^B D^A where B _* 0 is the matrix adjoint to B, are equivalent.     

Some observations on the Youla form and conjugate-normal matrices

Two square complex matrices A, B are said to be unitarily congruent if there is a unitary matrix U such that A = UBU^T. The Youla form is a canonical form under unitary congruence. We give a simple derivation of this form using coninvariant subspaces. For the special class of conjugate-normal matrices the associated Youla form is discussed.     

Exponential sums and rectangular partitions

Let F denote a finite field with q=p^f elements, and let σ(A) equal the trace of the square matrix A. This paper evaluates exponential sums of the form S(E,X₁,…,X_n)e{?σ(CX₁?X_nE)}, where S(E,X₁,…,X_n) denotes a summation over all matrices E,X₁,…,X_n of appropriate sizes over F, and C is a fixed matrix. This evaluation is then applied to the problem of counting ranked solutions to matrix equations of the form U₁?U_αA+DV₁?V_β=B where A,B,D are fixed matrices over F.     

Classification of pairs consisting of a linear and a semilinear map

L. Kronecker has found normal forms for pairs (A, B) of m-by-n matrices over a field F when the admissible transformations are of the type (A, B)→(SAT, SBT), where S and T are invertible matrices over F. For the details about these normal forms we refer to Gantmacher's book on matrices [5, Chapter XII]. See also Dickson's paper [3]. We treat here the following more general problem: Find the normal forms for pairs (A, B) of m-by-n matrices over a division ring D if the admissible transformations are of the type (A, B)→(SAT, SBJ(T)) where J is an automorphism of D. It is surprising that these normal forms (see Theorem 1) are as simple as in the classical case treated by Kronecker. The special case D=C, J=conjugation is essentially equivalent to the recent problem of Dlab and Ringel [4]. This is explained thoroughly in Sec. 6. We conclude with two open problems.     

On Some Ordering Properties of the Generalized Inverses of Nonnegative Definite Matrices

For any positive definite matrices A and B, it is known that A?B iff B^-1?A^-1. This paper investigates the extensions of the above result to any two real nonnegative definite matrices A and B.     

A note on “Constructing matrices with prescribed off-diagonal submatrix and invariant polynomials”

Let F be any field and let B a matrix of F^q×p. Zaballa found necessary and sufficient conditions for the existence of a matrix A=[A_ij]_i,j∈{1,2}∈F^(p+q)×(p+q) with prescribed similarity class and such that A₂₁=B. In an earlier paper [A. Borobia, R. Canogar, Constructing matrices with prescribed off-diagonal submatrix and invariant polynomials, Linear Algebra Appl. 424 (2007) 615-633] we obtained, for fields of characteristic different from 2, a finite step algorithm to construct A when it exists. In this short note we extend the algorithm to any field.     

A note on generating <Emphasis Type="Italic">P</Emphasis>-matrices

We prove that for any \({A,B\in\mathbb{R}^{n\times n}}\) such that each matrix S satisfying min(A, B) ≤ S ≤ max(A, B) is nonsingular, all four matrices A ^?1 B, AB ^?1, B ^?1 A and BA ^?1 are P-matrices. A practical method for generating P-matrices is drawn from this result.     

Involutory matrices over finite commutative rings

An n × n matrix A is called involutory iff A²=I_n, where I_n is the n × n identity matrix. This paper is concerned with involutory matrices over an arbitrary finite commutative ring R with identity and with the similarity relation among such matrices. In particular the authors seek a canonical set $\text{C}$ with respect to similarity for the n × n involutory matrices over R—i.e., a set $\text{C}$ of n × n involutory matrices over R with the property that each n × n involutory matrix over R is similar to exactly on matrix in $\text{C}$. Because of the structure of finite commutative rings and because of previous research, they are able to restrict their attention to finite local rings of characteristic a power of 2, and although their main result does not completely specify a canonical set $\text{C}$ for such a ring, it does solve the problem for a special class of rings and shows that a solution to the general case necessarily contains a solution to the classically unsolved problem of simultaneously bringing a sequence A₁,…,A_v of (not necessarily involutory) matrices over a finite field of characteristic 2 to canonical form (using the same similarity transformation on each A_i). (More generally, the authors observe that a theory of similarity fot matrices over an arbitrary local ring, such as the well-known rational canonical theory for matrices over a field, necessarily implies a solution to the simultaneous canonical form problem for matrices over a field.) In a final section they apply their results to find a canonical set for the involutory matrices over the ring of integers modulo 2^m and using this canonical set they are able to obtain a formula for the number of n × n involutory matrices over this ring.     

Jordan forms for mutually annihilating nilpotent pairs

In this paper we completely characterize all possible pairs of Jordan canonical forms for mutually annihilating nilpotent pairs, i.e. pairs (A,B) of nilpotent matrices such that AB=BA=0.     

Procrustes problems and associated approximation problems for matrices with k-involutory symmetries

We say that a matrix R∈C^n×n is k-involutary if its minimal polynomial is x^k-1 for some k?2, so R^k-1=R^-1 and the eigenvalues of R are 1,ζ,ζ²,…,ζ^k-1, where ζ=e^2πi/k. Let α,μ∈{0,1,…,k-1}. If R∈C^m×m, A∈C^m×n, S∈C^n×n and R and S are k-involutory, we say that A is (R,S,μ)-symmetric if RAS^-1=ζ^μA, and A is (R,S,α,μ)-symmetric if RAS^-α=ζ^μA.Let L be the class of m×n(R,S,μ)-symmetric matrices or the class of m×n(R,S,α,μ)-symmetric matrices. Given X∈C^n×t and B∈C^m×t, we characterize the matrices A in L that minimize ‖AX-B‖ (Frobenius norm), and, given an arbitrary W∈C^m×n, we find the unique matrix A∈L that minimizes both ‖AX-B‖ and ‖A-W‖. We also obtain necessary and sufficient conditions for existence of A∈L such that AX=B, and, assuming that the conditions are satisfied, characterize the set of all such A.     

The Lyapunov matrix equation SA+A1S=S1B1BS

The matrix equation $\text{SA+A}{}^1\text{S=S}{}^1\text{B}{}^1\text{BS}$ is studied, under the assumption that (A, B¹) is controllable, but allowing nonhermitian S. An inequality is given relating the dimensions of the eigenspaces of A and of the null space of S. In particular, if B has rank 1 and S is nonsingular, then S is hermitian, and the inertias of A and S are equal. Other inertial results are obtained, the role of the controllability of (A¹, B¹S¹) is studied, and a class of D-stable matrices is determined.     

Some explicit solutions of the additive Deligne-Simpson problem and their applications

In this paper we construct three infinite series and two extra triples (E₈ and ) of complex matrices B, C, and A=B+C of special spectral types associated to Simpson's classification in Amer. Math. Soc. Proc. 1 (1992) 157 and Magyar et al. classification in Adv. Math. 141 (1999) 97. This enables us to construct Fuchsian systems of differential equations which generalize the hypergeometric equation of Gauss-Riemann. In a sense, they are the closest relatives of the famous equation, because their triples of spectral flags have finitely many orbits for the diagonal action of the general linear group in the space of solutions. In all the cases except for E₈, we also explicitly construct scalar products such that A, B, and C are self-adjoint with respect to them. In the context of Fuchsian systems, these scalar products become monodromy invariant complex symmetric bilinear forms in the spaces of solutions.When the eigenvalues of A, B, and C are real, the matrices and the scalar products become real as well. We find inequalities on the eigenvalues of A, B, and C which make the scalar products positive-definite.As proved by Klyachko, spectra of three hermitian (or real symmetric) matrices B, C, and A=B+C form a polyhedral convex cone in the space of triple spectra. He also gave a recursive algorithm to generate inequalities describing the cone. The inequalities we obtain describe non-recursively some faces of the Klyachko cone.     

Interactions

Given a C^∗-algebra B, a closed ^*-subalgebra A⊆B, and a partial isometry S in B which interacts with A in the sense that S^∗aS=H(a)S^∗S and SaS^∗=V(a)SS^∗, where V and H are positive linear operators on A, we derive a few properties which V and H are forced to satisfy. Removing B and S from the picture we define an interaction as being a pair of maps (V,H) satisfying the derived properties. Starting with an abstract interaction (V,H) over a C^∗-algebra A we construct a C^∗-algebra B containing A and a partial isometry S whose interaction with A follows the above rules. We then discuss the possibility of constructing a covariance algebra from an interaction. This turns out to require a generalization of the notion of correspondences (also known as Pimsner bimodules) which we call a generalized correspondence. Such an object should be seen as an usual correspondence, except that the inner-products need not lie in the coefficient algebra. The covariance algebra is then defined using a natural generalization of Pimsner's construction of the celebrated Cuntz-Pimsner algebras.     

Semisimilarity for matrices over a division ring

Two square matrices A and B over a ring R are semisimilar, written A$\text{?}$B, if YAX=B and XBY=A for some (possibly rectangular) matrices X, Y over R. We show that if A and B have the same dimension, and if the ring is a division ring $\text{D}$, then A$\text{?}$B if and only if A² is similar to B² and rank(A^k)=rank(B^k), k=1,2,…     

Congruence and Conjunctivity of Matrices

Congruence of arbitrary square matrices over an arbitrary field is treated here by elementary classical methods, and likewise for conjunctivity of arbitrary square matrices over an arbitrary field with involution. Uniqueness results are emphasized, since they are largely neglected in the literature. In particular, it is shown that a matrix S is congruent [conjunctive] to S₀⊕S₁ with S₁ nonsingular, and that if S₁ here is of maximal size among all nonsingular matrices R₁ for which R₀⊕R₁ is congruent [conjunctive] to S, then the congruence [conjunctivity] class of S determines that of S₁. Partially canonical forms (most of them already known) are derived, to the extent that they do not depend on the field. Nearly canonical forms are derived for “neutral” matrices (those congruent or conjunctive with block matrices $\text{O}\text{N}\text{M}\text{O}$ with the two zero blocks being square). For a neutral matrix S over a field F,the F-congruence [F-conjunctivity] class of S is determined by the F-equivalence class of the pencil S+tS' [S+tS¹] and, if the pencil is nonsingular, by the F[t]-equivalence class of S+tS' [S+tS¹].     

On the nonsingular symmetric factors of a real matrix

For a given real square matrix A this paper describes the following matrices: (¹) all nonsingular real symmetric (r.s.) matrices S such that A = S^?1T for some symmetric matrix T.All the signatures (defined as the absolute value of the difference of the number of positive eigenvalues and the number of negative eigenvalues) possible for feasible S in (¹) can be derived from the real Jordan normal form of A. In particular, for any A there is always a nonsingular r.s. matrix S with signature S ? 1 such that A = S^?1T.     

Pairs of alternating forms and products of two skew-symmetric matrices

Let k be a field, and let S,T,S₁,T₁ be skew-symmetric matrices over k with S,S₁ both nonsingular (if k has characteristic 2, a skew-symmetric matrix is a symmetric one with zero diagonal). It is shown that there exists a nonsingular matrix P over k with P'SP = S₁, P'TP = T₁ (where P' denotes the transpose of P) if and only if S^-1T and S^-1₁T₁ are similar. It is also shown that a 2m×2m matrix over k can be factored as ST, with S,T skew-symmetric and S nonsingular, if and only if A is similar to a matrix direct sum B⊕B where B is an m×m matrix over k. This is equivalent to saying that all elementary divisors of A occur with even multiplicity. An extension of this result giving necessary and sufficient conditions for a square matrix to be so expressible, without assuming that either S or T is nonsingular, is included.