A result on integral symmetric matrices

Let A be an integral matrix such that det A = 1 mod mA ≡ A^T mod m, where m is odd. It is shown that a symmetric integral matrix B of determinant 1 exists such that B ≡ A mod m. The result is false if m is even.     

General polynomial roots and their multiplicities in O(N)memory and O(N2)Time

For a given real or complex polynomial p of degree n we modify the Euclidean algorithm to find a general tridiagonal matrix representation T of the monic version of p and then use the tridiagonal DQR eigenvalue algorithm on T in order to find

all roots ofp with their multiplicities in O(n²) operations

and 0(n) storage. We include details of the implementation and comparisons with several, standard and recent, essentially 0(n³) polynomial root finders.     

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.     

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.     

Subpermanents of doubly stochastic matrices

It is shown that if all subpermaneats of order k of an n × n doubly stochastic matrix are equal for some k ≤ n - 2, then all the entries of the matrix must be equal to 1/n.     

An approximate, multivariable version of Specht's theorem

In this article we provide generalizations of Specht's theorem which states that two n × n matrices A and B are unitarily equivalent if and only if all traces of words in two non-commuting variables applied to the pairs (A, A*) and (B, B*) coincide. First, we obtain conditions which allow us to extend this to simultaneous similarity or unitary equivalence of families of operators, and secondly, we show that it suffices to consider a more restricted family of functions when comparing traces. Our results do not require the traces of words in (A, A*) and (B, B*) to coincide, but only to be close.     

Linear preservers of matrices of rank-2

Let T be a linear operator on the space of all m×n matrices over any field. we prove that if T maps rank-2 matrices to rank-2 matrices then there exist nonsingular matrices U and V such that either T(X)=UXV for all matrices X, or m=n and T(X)=UX^tV for all matrices X where X^t denotes the transpose of X.     

Zero-term rank preservers

We obtain characterizations of those linear operators that preserve zero-term rank on the m×n matrices over antinegative semirings. That is, a linear operator T preserves zero-term rank if and only if it has the form T(X)=P(B∘X)Q, where P, Q are permutation matrices and B∘X is the Schur product with B whose entries are all nonzero and not zero-divisors.     

Manifolds of idemopotent matrices

Let M_n be the set of n×n matrices and r a nonnegative integer with r ≤ n. It is known,from Lie groups, that the rank r idempotent matrices in M_n form an arcwise connected 2n (n-r)-dimensional analytic manifold. This paper provides an elementary proof of this result making it accessible to a larger audience.     

An extension of flanders theorem to several matrices

The Flanders Theorem relates the matrices AB and BA and provides a necessary and sufficient condition for the consistency of the matrix system P = ABQ = BA In this paper, we generalize the Flanders condition for several matrices.     

Permanents of doubly stochastic matrices with fixed zero pattern

The doubly stochastic matrices with a given zero pattern which are closest in Euclidean norm to J_nn, the matrix with each entry equal to 1/n, are identified. If the permanent is restricted to matrices having a given zero pattern confined to one row or to one column, the permanent achieves a local minimum at those matrices with that zero pattern which are closest to J_nn. This need no longer be true if the zeros lie in more than one row or column.     

Large spaces of symmetric matrices of bounded rank are decomposable

Let k and n be positive integers such that k≤n. Let S_n(F) denote the space of all n×n symmetric matrices over the field F with char F≠2. A subspace L of S_n(F) is said to be a k-subspace if rank A≤k for every AεL.

Now suppose that k is even, and write k=2r. We say a k∥-subspace of S_n(F) is decomposable if there exists in Fⁿ a subspace W of dimension n-r such that x^tAx=0 for every xεWAεL.

We show here, under some mild assumptions on kn and F, that every k∥-subspace of S_n(F) of sufficiently large dimension must be decomposable. This is an analogue of a result obtained by Atkinson and Lloyd for corresponding subspaces of F^m,n.     

Some involutory similarities

Matrices A,B over an arbitrary field F, when given to be similar to each other, are shown to be involutorily similar (over F) to each other (i.e.B = CAC^-1for some C = C^-1over F) in the following cases: (1)B= aI - Afor some a ε F and (2) B = A^-1. Result (2) for the cases where char F ≠ 2 is essentially a 1966 result of Wonenburger.     

The minimal polynomial of a matrix

Let Rbe a finite dimensional central simple algebra over a field FA be any n× n matrix over R. By using the method of matrix representation, this paper obtains the structure formula of the minimal polynomial q_A(λ) of A over F. By using q_A(λ), this paper discusses the structure of right (left) eigenvalues set of A, and obtains the necessary and sufficient condition that a matrix over a finite dimensional central division algebra is similar to a diagonal matrix.     

Positive and real-positive solutions to the equation axa*=c in C*-algebras

We obtain new representations for the general positive and real-positive solutions of the equation axa*=c in a C*-algebra using the characterization of positivity based on a matrix representation of an element and the generalized Schur complement. Applications to the equation AXA*=C for operators between Hilbert spaces and for finite matrices are given.     

The doubly stochastic matrix equations x:aty= A and X Aty=t

For a doubly stochastic matrix A, each of the equations x:a^ty= A and X A^ty=^t is shown to have doubly stochastic solutions X and Y if and only if A lies in a subgroup of the semigroup of all doubly stochastic matrices of a given order. All elements of this semigroup which are left regular, right regular, or intra-regular are identified.     

Spectrally arbitrary zero-nonzero patterns of order 4

In this article the authors characterize all the 4×4 zero-nonzero patterns that are spectrally arbitrary. Several observations and conjectures are presented for the n × n case.     

Matrices that preserve vectors of fixed sign variation

For each k≥ 0, those nonsingular matrices that transform the set of totally nonzero vectors with k sign variations into (respectively, onto) itself are studied. Necessary and sufficient conditions are provided. The cases k=0,1,2,n-3,n-2,n-1 are completely characterized.     

Discriminant preserving linear maps

Consider the n-square matrices over an infiniie field Kas an n²-dimcnsional vector space M( nK). We determine all linear maps Ton M(nK) such that discriminant TX- discriminant Xfor all Xin M(nK)