Composition of binary integral quadratic forms via integral 2 × 2 matrices and composition of matrix classes

A treatment by integral matrices is given for composition of pairs of integral quadratic forms with the same discriminant. The forms are associated with a pair of similar 2 × 2 matrices AB with irrational eigen values which generate the maximal order. The most general integral similarity between AB is given by a matrix whose entries are linear forms in two indeterminates with integral coefficients. This matrix is a "compositum" of two factors of the same nature. By equating determinants a composition of two quadratic forms results. The method can be generalized to n × n matrices.     

Linear operators preserving unitarily invariant norms of matrices

Let F_{m × n} be the set of all m × n matrices over the field F = C or R Denote by U_n(F) the group of all n × n unitary or orthogonal matrices according as F = C or F-R. A norm N() on Fm ×n, is unitarily invariant if N(UAV) = N(A): for all A ∈ F _m×n U ∈ U_m(F). and V ∈ U_n(F). We characterize those linear operators TF_{m × n} → F_{m × n}which satisfy N (T(A)) = N(A)for all A ∈ F_{m × n}

for a given unitarily invariant norm N(). It is shown that the problem is equivalent to characterizing those operators which preserve certain subsets in F_{m × n} To develop the theory we prove some results concerning unitary operators on F_{m × n} which are of independent interest.     

The rank of the difference of matrices with prescribed similarity classes

Let F be a field and let A,B be n × n matrices over I. We study the rank of A' - B' when A and B run over the set of matrices similar to A and B, respectively.     

Linear operators that preserve pairs of matrices which satisfy extreme rank properties

A pair of m×n matrices (A,B) is called rank-sum-maximal if rank(A+B)=rank(A)+rank(B), and rank-sum-minimal if rank(A+B)=|rank(A)−rank(B)|. We characterize the linear operators that preserve the set of rank-sum-minimal matrix pairs, and the linear operators that preserve the set of rank-sum-maximal matrix pairs over any field with at least min(m,n)+2 elements and of characteristic not 2.     

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.     

Mappings preserving spectrum and commutativity on Hermitian matrices

The general form of a continuous mapping φ acting on the real vector space of all n × n complex Hermitian or real symmetric matrices, and preserving spectrum and commutativity, is derived. It turns out that φ is either linear or its image forms a commutative set.     

Generalized equivalence of pairs of matrices

Pairs (A₁B₁) and (A₂B₂) of matrices over a principal ideal domain R are called the generalized equivalent pairs if A₂=UA₁V₁B₂=UB₁V₂ for some invertible matrices UV₁V₂ over R. A special form is established to which a pair of matrices can be reduced by means of generalized equivalent transformations. Besides necessary and sufficient conditions are found, under which a pair of matrices is generalized equivalent to a pair of diagonal matrices. Applications are made to study the divisibility of matrices and multiplicative property of the Smith normal form.     

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.     

On the jordan form of the tensor product over fields of prime characteristic

Let A, B denote the companion matrices of the polynomials x^m,xⁿ over a field F of prime order p and let λ,μ be non-zero elements of an extension field K of F. The Jordan form of the tensor product (λI + A)⊗(μI + B) of invertible Jordan matrices over K is determined via an equivalent study of the nilpotent tranformation S of m × n matrices X over F where(X)S = A ^TX + XB. Using module-theoretic concepts a Jordan basis for S is specified recursively in terms of the representations of m and n in the scale of p, and reduction formulae for the elementary divisors of S are established.     

Geometry of 2×2 Hermitian matrices II

Let Z be a field of characteristic ≠2, D be a quaternion division algebra over Z and have a nonstandard involution of the first kind. The fundamental theorem of geometry of 2× 2 Hermitian matrices over D are proved. Thus, if D is a quaternion division algebra over Z with an involution of the first kind, then the fundamental theorem of geometry of 2× 2 Hermitian matrices over D are obtained.     

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.     

A comparison of nonnegative real ranks and their preservers

This paper concerns three notions of rank of matrices over semirings; real rank, semiring rank and column rank. These three rank functions are the same over subfields of reals but differ for matrices over subsemirings of nonnegative reals. We investigate the largest values of r for which the real rank and semiring rank, real rank and column rank of all m×n matrices over a given semiring are both r, respectively. We also characterize the linear operators which preserve the column rank of matrices over certain subsemirings of the nonnegative reals.     

A note on rank-additivity

A new criterion is given for rank additivity of a collection of m × n complex matrices.     

Characteristic roots and vectors of a diifferentiable family of symmetric matrices

LetA(x) be a differentiable family of k × k symmetric matrices where x runs through a domain D in RⁿWe prove that if λ is a continuous function onDsuch that, for every x εD,λ(x) is a characteristic root of A(x) of constant multiplicity m, then λ is a differentiable function and there exists, locally, a differentiable family of ortho-normal bases for the eigenspace. The case n = 1 has been known in the standard treatises on the perturbation theory for linear operators.     

Commutativity preserving linear maps and Lie automorphisms of strictly triangular matrix space

In this paper we classify linear maps preserving commutativity in both directions on the space N(F) of strictly upper triangular (n+1)×(n+1) matrices over a field F. We show that for n3 a linear map on N(F) preserves commutativity in both directions if and only if =^′+f where ^′ is a product of standard maps on N(F) and f is a linear map of N(F) into its center.     

Examples of constant rank spaces

The maximum dimension of a space of (k+1)×(k4+s-1) complex matrices of rank k is either s or s+1. Only when s divides k is it possible for the maximum to be s+1. This much is known. In this paper we produce for each k, a multiple of s, an (s+l)-dimensional space of (k+1)×(k+s-1) complex matrices whose non-zero members all have rank k. In the notation introduced by Sylvester l(k,k+1,k+s-1)=s+1 whenever s divides k.     

The center of the ring of 3×3 generic matrices

Following Procesi, the center of the division ring of generic matrices over a field F is described as the fixed field of the symmetric group acting on a purely transcendental extension of F. For 3×3 matrices, the center is shown to be purely transcendental over F. In characteristic zero this is equivalent to saying that the field of simultaneous rational invariants of 3×3 matrices over F is a purely transcendental extension field of F.     

An extremal problem for positive defintie matrices

A problem studied by Flanders (1975) is minimize the function f(R)=tr(SR+TR^-1) over the set of positive definite matrices R, where S and T are positive semi-definite matrices. Alternative proofs that may have some intrinsic interest are provided. The proofs explicitly yield the infimum of f(R). One proof is based on a convexity argument and the other on a sequence of reductions to a univariate problem.     

Linear transformations on matrices: The invariance of commuting pairs of matrices

Let L be a linear transformation on the set of all n×n matrices over an algebraically closed field of characteristic 0. It is shown that if AB=BA implies L(A)L(B)=L(B)L(A) and if either L is nonsingular or the implication in the hypothesis can also be reversed, then L is a sum of a scalar multiple of a similarity transformation and a linear functional times the identity transformation.