On matrices stochastically similar to matrices with equal diagonal elements

A necessary and sufficient condition for a matrix to be stochastically similar to a matrix with equal diagonal elements is obtained Aand B are called Stochastically similar if B=SAS^{- 1} where S is quasi-stochastic i.e., all row sums of .S are I. An inverse elementary divisor problem for quasi-stochastic matrices is also considered.     

Linear Operators that Preserve Commuting Pairs of Nonnegative Real Matrices

A pair of n×n matrices (A, B) is called a commuting pair if AB=BA. We characterize the linear operators that preserve the set of commuting pairs of matrices over a subsemiring of nonnegative real numbers.     

Strong 3-commutativity preserving maps on standard operator algebras

Let X be a Banach space of dimension ≥ 2 over the real or complex field F and A a standard operator algebra in B(X). A map Φ :A →A is said to be strong 3-commutativity preserving if [Φ(A), Φ(B)]3 = [A,B]3 for all A,B∈ A, where[A,B]3 is the 3-commutator of A,B defined by[A, B]3 = [[[A, B],B],B] with [A,B] = AB-BA. The main result in this paper is shown that.,if Φ is a surjective map on A, then Φ is strong 3-commutativity preserving if and only if there exist a functional h : A →F and a scalar λ∈ F with λ~4 = 1 such that Φ(A)=λ A+h(A)I for all A ∈ A.     

Additive Maps Preserving Nilpotent Perturbation of Scalars

Let X be a Banach space over F(= R or C) with dimension greater than 2. Let N(X) be the set of all nilpotent operators and B_0(X) the set spanned by N(X). We give a structure result to the additive maps on FI + B_0(X) that preserve rank-1 perturbation of scalars in both directions. Based on it, a characterization of surjective additive maps on FI + B_0(X) that preserve nilpotent perturbation of scalars in both directions are obtained. Such a map Φ has the form either Φ(T) = cAT A~(-1)+ φ(T)I for all T ∈ FI + B_0(X) or Φ(T) = cAT*A~(-1)+ φ(T)I for all T ∈ FI + B_0(X), where c is a nonzero scalar,A is a τ-linear bijective transformation for some automorphism τ of F and φ is an additive functional.In addition, if dim X = ∞, then A is in fact a linear or conjugate linear invertible bounded operator.     

Linear operators preserving unitary t-congruence (orthogonal similarity) on complex (real) matrices

Two complex (real) square matrices A and B are said io be unitarily t-congruent (orthogonally similar) it there exists a unitary (an orthogonal) matrix U such that A=UBU¹ We characterize those linear operators that preserve unitary t-congruence on complex matrices and those linear operators that preserve orthogonal similarity on real matrices. This answers a question raised in a paper by Y. P. Hong, R. A. Horn and the first author.     

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.     

On the normality of operator products

If A and B^* are hyponormal operators such that AB is compact, then the normality of AB implics that of BA. This is a generalization of an old result of Wiegmann and Kaplansky on products of normal operators.     

Some inequalities for commutators and an application to spectral variation. II

Inequalities that compare unitarily invariant norms of A - B and those of AΓ - ΓB and Γ^-1A - B Γ^-1 are obtained, where both A and B are either Hermitian or unitary or normal operators and Γ is a positive definite operator in a complex separable Hilbert space. These inequalities are then applied to derive bounds for spectral variation of diagonalisable matrices. Our new bounds improve substantially previously published bounds.     

An inequality for M-matrices

If AB are n × n M matrices with dominant principal diagonal, we show that 3[det(A + B)]^1/n ≥ (det A)^1/n + (det B)^1/n.     

Convergent regular splittings for nonnegative matrices

MP matrices are those real matrices which possess a nonnegative, nonsingular l-inverse. This paper characterizes the nonnegative MP matrices and hence, determines when a nonnegative matrix A has a convergent regular splitting M—Q which induces the linear stationary iterative scheme x_k+1=M^-1Qx_k+M^-1b to solve Ax=b.     

Direct fail-proof triangularization algorithms for AX + XB = C with Error-Free and parallel implementations

Two 0(mn³) inversion-free direct algorithms to compute a solution of the linear system AX +XB = C by triangularizing a Hessenberg matrix are presented. Without any loss of generality the matrix A is assumed upper Hessenberg and the order m of A the order n of B. The algorithms have an in-built consistency check, are capable of pruning redundant rows and converting the resulting matrix into a full row rank matrix, and permit A and —B to be any square matrices with common or distinct eigenvalues. In addition, these algorithms can also solve the homogeneous system AX +XB = 0 (null matrix C). An error-free implementation of the solution X using multiple modulus residue arithmetic as well as a parallelization of the algorithms is discussed.     

A theory of electromagnetic and gravitational fields

Careful calculations using classical field theory show that if a macroscopic ball with uniform surface charge (say, a billiard ball with 1E6 excess electrons) is released near the surface of the earth, it will almost instantaneously accelerate to relativistic speed and blow a hole in the ground. This absurd prediction is just the macroscopic version of the self-force problem for charged particles [1]. Furthermore, if one attempts to develop from electromagnetism a parallel theory for gravitational [2], the result is the same, self-acceleration.

The basis of the new theory is a measure of energy density for any wave equation [3–5]. Given any solution of any four-vector wave equation in spacetime (for example, the potentials (c^-1φA)=(A⁰,A¹,A²,A³) in electromagnetism), one can form the 16th first order partial derivatives of the vector components, with respect to the time and space variables (ct,x) = (x⁰, x¹, x², x³). The sum of the squares of the 16 terms is a natural energy function [6, p. 283] (satisfying a conservation law . Such energy functions are routinely utilized by mathematicians as Lyapunov functions in the theory of stability of waves with boundary conditions. A Lagrangian using this sum leads to a new energy tensor for electromagnetic and gravitational fields, an alternative to that in [7].     

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.     

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.     

Cases of equality for the spectral norm submultiplicativity of the hadamard product

In this note, we characterize those pairs of nonzero r-by-d complex matrices that satisfy N₂(A ∘ B) = N₂(A)N₂(B), in which N₂(·) is the spectral norm and · is the Hadamard product.     

Convex sets of schur stable and stable matrices

We derive characterizations for the Schur stability and the stability of all convex combinations of kk≥2, given real square matrices. Moreover, we characterize these properties for the set r(A, B) (c(A,B), resp.) of square matrices whose rows (columns, resp.) are independent convex combinations of the rows (columns, resp.) of two real matrices A and B. Our results can be viewed as contributions to the problem of robustness of matrix properties. This paper continues our paper [4].     

On real matrices with positive definite symmetric component

Inequalities concerning real square matrices A with positive definite symmetric component A+A^*are derived from certain inertia relations which hold for any complex (not necessarily real) square matrices A with positive definite

A+A^*     

On generalized Boolean functions II

This cycle of papers is based on the concept of generalized Bolean functions introduced by the author in the first article of the series. Every generalized Boolean function f:Bⁿ→B can be written in a manner similar to the canonical disjunctive form using some function defined on A×B, where A is a finite subset of B containing 0 and 1. The set of those functions f is denoted by GBF_n[A]. In this paper the following questions are presented: (1) What is the relationship between GBF_n[A₁] and GBF_n[A₂] when A₁A₂. (2) What can be said about GBF_n[A₁∩A₂] and GBF_n[A₁A₂] in comparison with GBF_n[A₁]∩GBF_n[A₂] and GBF_n[A₁]GBF_n[A₂], respectively.     

The construction of irreducible normalized representations of the general linear group

A method is described for constructing in an explicit form an irreducible representation T of M_n(F), the set of all n × n matrices over the real or complex field F, satisfying the condition T(A^*)=T^*(A) for all A∈M_n(F).