Some results on symplectic matrices

The principal results are that if A is an integral matrix such that AA^T is symplectic then A = CQ, where Q is a permutation matrix and C is symplectic; and that if A is a hermitian positive definite matrix which is symplectic, and B is the unique hermitian positive definite pth.root of A, where p is a positive integer, then B is also symplectic.     

Some results on symplectic matrices

The principal results are that if A is an integral matrix such that AA^T is symplectic then A = CQ, where Q is a permutation matrix and C is symplectic; and that if A is a hermitian positive definite matrix which is symplectic, and B is the unique hermitian positive definite pth.root of A, where p is a positive integer, then B is also symplectic.     

A note on matrix equivalence

Let R be a principal ideal ringR_n the ring of n × n it matrices over R. It is shown that if A, B, X, Y are elements of R* such that A = XB, B = YA, then A and B are left equivalent. Some consequences are given.     

Absolutely indecomposable symmetric matrices

Let A be a symmetric matrix of size n×n with entries in some (commutative) field K. We study the possibility of decomposing A into two blocks by conjugation by an orthogonal matrix T∈Mat_n(K). We say that A is absolutely indecomposable if it is indecomposable over every extension of the base field. If K is formally real then every symmetric matrix A diagonalizes orthogonally over the real closure of K. Assume that K is a not formally real and of level s. We prove that in Mat_n(K) there exist symmetric, absolutely indecomposable matrices iff n is congruent to 0, 1 or −1 modulo 2s.     

Additive mappings on symmetric matrices

Additive mappings, which do not increase the minimal rank of symmetric matrices are classified in characteristic two or three.     

A min-max theorem for complex symmetric matrices

We optimize the form Re x^tTx to obtain the singular values of a complex symmetric matrix T. We prove that for ,

     

Products of idempotent matrices over integral domains

We say that a ring R has the idempotent matrices property if every square singular matrix over R is a product of idempotent matrices. It is known that every field, and more generally, every Euclidean domain has the idempotent matrices property. In this paper we show that not every integral domain has the idempotent matrices property and that if a projective free ring has the idempotent matrices property then it must be a Bezout domain. We also show that a principal ideal domain has the idempotent matrices property if and only if every fraction a/b with b≠0 has a finite continued fraction expansion. New proofs are also provided for the results that every field and every Euclidean domain have the idempotent matrices property.     

The scrambling index of symmetric primitive matrices

A nonnegative square matrix A is primitive if some power A^k>0 (that is, A^k is entrywise positive). The least such k is called the exponent of A. In [2], Akelbek and Kirkland defined the scrambling index of a primitive matrix A, which is the smallest positive integer k such that any two rows of A^k have at least one positive element in a coincident position. In this paper, we give a relation between the scrambling index and the exponent for symmetric primitive matrices, and determine the scrambling index set for the class of symmetric primitive matrices. We also characterize completely the symmetric primitive matrices in this class such that the scrambling index is equal to the maximum value.     

Matrices which commute with doubly stochastic matrices

The set doubly stochastic matrices which commute with the doubly stochastic matrices of any particular given rank is determined.     

Characteristic polynomials of symmetric matrices III: some counterexamples

When is a monic polynomial the characteristic polynomial of a symmetric matrix over an integral domain D? Known necessary conditions are shown to be insufficient when D is the field of 2-adic numbers and when D is the rational integers. The latter counterexamples lead to totally real cubic extensions of the rationals whose difierents are not narrowly equivalent to squares. Furthermorex³-4x+1 is the characteristic polynomial of a rational symmetric matrix and is the characteristic polynomial of an integral symmetric p-adic matrix for every prime p, but is not the characteristic polynomial of a rational integral symmetric matrix.     

An eigenvalue localization theorem for pentadiagonal symmetric Toeplitz matrices

We express the eigenvalues of a pentadiagonal symmetric Toeplitz matrix as the zeros of explicitly given rational functions.     

Matrices with maximum kth local exponent in the class of doubly symmetric primitive matrices

Let A be a primitive matrix of order n, and let k be an integer with 1?k?n. The kth local exponent of A, is the smallest power of A for which there are k rows with no zero entry. We have recently obtained the maximum value for the kth local exponent of doubly symmetric primitive matrices of order n with 1?k?n. In this paper, we use the graph theoretical method to give a complete characterization of those doubly symmetric primitive matrices whose kth local exponent actually attain the maximum value.     

Factorization of singular integer matrices

It is well known that a singular integer matrix can be factorized into a product of integer idempotent matrices. In this paper, we prove that every n  × n (n > 2) singular integer matrix can be written as a product of 3n + 1 integer idempotent matrices. This theorem has some application in the field of synthesizing VLSI arrays and systolic arrays.     

Rank properties of subspaces of symmetric and hermitian matrices over finite fields

We investigate constant rank subspaces of symmetric and hermitian matrices over finite fields, using a double counting method related to the number of common zeros of the corresponding subspaces of symmetric bilinear and hermitian forms. We obtain optimal bounds for the dimensions of constant rank subspaces of hermitian matrices, and good bounds for the dimensions of subspaces of symmetric and hermitian matrices whose non-zero elements all have odd rank.     

On the eigenvalue problem for a particular class of finite Jacobi matrices

A function f with simple and nice algebraic properties is defined on a subset of the space of complex sequences. Some special functions are expressible in terms of f, first of all the Bessel functions of first kind. A compact formula in terms of the function f is given for the determinant of a Jacobi matrix. Further we focus on the particular class of Jacobi matrices of odd dimension whose parallels to the diagonal are constant and whose diagonal depends linearly on the index. A formula is derived for the characteristic function. Yet another formula is presented in which the characteristic function is expressed in terms of the function f in a simple and compact manner. A special basis is constructed in which the Jacobi matrix becomes a sum of a diagonal matrix and a rank-one matrix operator. A vector-valued function on the complex plain is constructed having the property that its values on spectral points of the Jacobi matrix are equal to corresponding eigenvectors.     

Generalized Bowen-Franks groups of integral matrices with the same zeta function

In this paper the relation between the zeta function of an integral matrix and its generalized Bowen-Franks groups is studied. Suppose that A and B are nonnegative integral matrices whose invertible part is diagonalizable over the field of complex numbers and A and B have the same zeta function. Then there is an integer m, which depends only on the zeta function, such that, for any prime q such that gcd(q,m)=1, for any g(x)∈Z[x] with g(0)=1, the q-Sylow subgroup of the generalized Bowen-Franks group BF_g(x)(A) and BF_g(x)(B) are the same. In particular, if m=1, then zeta function determines generalized Bowen-Franks groups.     

Strong linear preservers of rank reverse permutability on triangular matrices

Let F be a field with ∣F∣ > 2 and T_n(F) be the set of all n × n upper triangular matrices, where n ? 2. Let k ? 2 be a given integer. A k-tuple of matrices A₁, …, A_k ∈ T_n(F) is called rank reverse permutable if rank(A₁ A₂ ? A_k) = rank(A_k A_k−1 ? A₁). We characterize the linear maps on T_n(F) that strongly preserve the set of rank reverse permutable matrix k-tuples.     

Subspaces of matrices with special rank properties

Let K be a field and let M_m×n(K) denote the space of m×n matrices over K. We investigate properties of a subspace M of M_m×n(K) of dimension n(m-r+1) in which each non-zero element of M has rank at least r and enumerate the number of elements of a given rank in M when K is finite. We also provide an upper bound for the dimension of a constant rank r subspace of M_m×n(K) when K is finite and give non-trivial examples to show that our bound is optimal in some cases. We include a similar a bound for the maximum dimension of a constant rank subspace of skew-symmetric matrices over a finite field.