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.     

The maximun of i(XAX-1+B)

Let F be an algebraically closed field. We denote by i(A) the number of invariant polynomials of a square matrix A, which are different from 1. For A,B any n×n matrices over F, we calculate the maximum of i(XAX^-1+B), where X runs over the set of all non-singular n×n matrices over F.     

A matrix problem associated with computing Cohen-Macaulay type

Suppose A∈M_n×m(F), B∈M_n×t(F) for some field F. Define Г(AB) to be the set of n×n diagonal matrices D such that the column space of DA is contained in the column space of B. In this paper we determine dim Г(AB). For matrices AB of the same rank we provide an algorithm for computing dim Г(AB).     

The Best Lower Bound for the Permanent of a Correlation Matrix of Rank Two

If 1≤k≤n, then Cor(n,k) denotes the set of all n×n real correlation matrices of rank not exceeding k. Grone and Pierce have shown that if A∈Cor (n, n-1), then per(A)≥n/(n-1). We show that if A∈Cor(n,2), then , and that this inequality is the best possible.     

On the permanent of doubly stochastic matrices with zero diagonal

The permanent function on the set of n×n doubly stochastic matrices with zero main diagonaln≤4, attains its minimum uniquely at the matrix whose off-diagonal entries are all equal to l/(n-1).     

Two generators for the general linear groups over finite fields

For every finite field F and every n≥2, the group GL(n,F) can be generated by two elements (which are explicitly described). The multiplicative semigroup of all n by n matrices over F can then be generated by three elements.     

Multiplicative properties of generalized matrix functions

We consider scalar-valued matrix functions for n×n matrices A=(a_ij) defined by Where G is a subgroup of S_n the group of permutations on n letters, and χ is a linear character of G. Two such functions are the permanent and the determinant. A function (1) is multiplicative on a semigroup S of n×n matrices if d(AB)=d(A)d(B) AB∈S.

With mild restrictions on the underlying scalar ring we show that every element of a semigroup containing the diagonal matrices on which (1) is multiplicative can have at most one nonzero diagonal(i.e., diagonal with all nonzero entries)and conversely, provided that χ is the principal character(χ≡1).     

Products of complic cosquares and pseudo-involutory matrices

LetF be a field with (nontrivial) involution (i.e.F-conjugation). A nonsingular matrix Aover Fis called a complic F-cosquare provided A=S^*-1for some matrix Sover Fand is called p.i. (pseudo-involutory) provided A=A^-1 It is shown that Ais a complic F-cosquare iff Ais the product of two p.i. matrices over Fand that det (AA)=1 iff Ais the product of two complic F-cosquares (hence iff A is the product of four p.i. matrices over F). It is conjectured that, except for one obvious case (2 x 2 matrices over the field of order 2), every unimodular matrix A over an arbitrary field Fis a product S¹ST:1T with S1 and Tover FThis conjecture is proved for matricesAof order ≤3.     

Alternative proofs of a theorem of Moyls and Marcus on the numerical range of a square matrix

Moyls and Marcus [4] showed that for n≤4,n×n an complex matrix A is normal if and only if the numerical range of A is the convex hull of the eigenvalues of A. When n≥5, there exist matrices which are not normal, but such that the numerical range is still the convex hull of the eigenvalues. Two alternative proofs of this fact are given. One proof uses the known structure of the numerical range of a 2×2 matrix. The other relies on a theorem of Motzkin and Taussky stating that a pair of Hermitian matrices with property L must commute.     

Products of involutory matrices. I

It is shown that, for every integer ?1 and every field F, each n×n matrix over F of determinant ±1 is the product of four involutory matrices over F. Products of three ×n involutory matrices over F are characterized for the special cases where n?4 or F has prime order ?5. It is also shown for every field F that every matrix over F of determinant ±1 having no more than two nontrivial invariant factors is a product of three involutory matrices over F.     

A result about determinantal divisors

Let Rbe a principal ideal ringR_n the ring of n× nmatrices over R, and d_k(A) the kth determinantal divisor of Afor 1 ≤ k≤ n, where Ais any element of R_n, It is shown that if A,BεR_n, det(A) det(B:) ≠ 0, then d_k(AB) ≡ 0 mod d_k(A) d_k(B). If in addition (det(A), det(B)) = 1, then it is also shown that d_k(AB) = d_k(A) d_k(B). This provides a new proof of the multiplicativity of the Smith normal form for matrices with relatively prime determinants.     

Linear mappings which are rank-k nonincreasing, II

We prove the following result. Let F be an infinite field of characteristic other than two. Let k be a positive integer. Let S_n(F) denote the space of all n × n symmetric matrices with entries in F, and let T:S_n(F)→S_n(F) be a linear operator. Suppose that T is rank-k nonincreasing and its image contains a matrix with rank higher than K. Then, there exist λεF and PεF^n,n such that T(A)=λPAP^t for all AεS_n(F). λ can be chosen to be 1 if F is algebraically closed and ±1 if F=R, the real field.     

Linear transformations on symmetric matrices II

Let Tbe a linear mapping on the space of n× nsymmetric matrices over a field Fof characteristic not equal to two. We obtain the structure of Tfor the following cases:(i) Tpreserves matrices of rank less than three; (ii) Tpreserves nonzero matrices of rank less than K + 1 where Kis a fixed positive integer less than nand Fis algebraically closed; (iii) Tpreserves rank Kmatrices where Kis a fixed odd integer and Fis algebraically closed.     

Finding all essential terms of a characteristic maxpolynomial

Let us denote ab=max(a,b) and ab=a+b for and extend this pair of operations to matrices and vectors in the same way as in linear algebra. We present an O(n²(m+n log n)) algorithm for finding all essential terms of the max-algebraic characteristic polynomial of an n×n matrix over with m finite elements. In the cases when all terms are essential, this algorithm also solves the following problem: Given an n×n matrix A and k{1,…,n}, find a k×k principal submatrix of A whose assignment problem value is maximum.     

The identity as a matrix commutator

If A is an n×n matrix over a field F of positive characteristic p, then I_n=AB-BA, for some matrix B, iff p divides the size of each Jordan block of A.     

Matrices as the sum of four squares

For each n≥2, every integral n×n matrix is the sum of at most four squares.     

A note on commuting pairs of matrices

We give a short proof of the Motzkin-Taussky result that the variety of commuting pairs of matrices is irreducible. An easy consequence of this is that any two generated commutative subalgebra of n×n matrices has dimension at most n. We also answer an old question of Gerstenhaber by showing that the set of commuting triples of n×n matrices is not irreducible for n≥32.     

On k-spaces of real matrices

It is shown that if W is a linear subspace of real n × n matrices, such that rank (A) = k for all 0 ≠ A ∈ W, then dim W≤ n. If dim W = n.5≤ n is prime, and 2 is primitive modulo n then k =1.     

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.     

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.