Solvability of the Matric Equation AX = B

Let A be an n×s matrix of rank r, B be an n×t matrix of rank ρ?r, and X be an s×t matrix. This paper discusses conditions on the matrices A and B so that the matric equation AX=B will have solutions for the matrix X.     

Ranked solutions of AXC=B and AX=B

Let GF(pⁿ) denote the finite field of pⁿ elements, p odd. Let A be an s×m matrix of rank ?, B be an s×t matrix of rank β, and C be an f×t matrix of rank v. This paper discusses the number of m×f matrices X of rank k over GF(pⁿ) which are solutions to the matric equations AXC=B or AX=B.     

Controllability,observability and the solution of AX - XB = C

A closed-form finite series representation of the unique solution X of the matrix equation AX ? XB=C is developed. Using this representation, the image, kernel, and rank of X are related to the controllable and unobservable subspaces of the (A, C) and (C, B) pairs respectively. Bounds on the rank of X are obtainedin terms of the dimensions of these subspaces. In the case that C has unitary rank, an exact calculation of rank X is made. The generic rank of X with A, B fixed and C generic is evaluated.     

Properties of the {ell}1-solutions of the Linear Matrix Equation AX+YB=C

The properties of the ₁ of the linear matrix equation AX+YB=Care investigated, where A, B, and C are given real matricesof dimensions m x r, s x n, and m x n, respectively, with m> r and n > s. An algorithm which is an ₁version of thegeneral alternating method is developed. This algorithm utilizesa special form of the equation AX+YB=C and adjusts X and Y alternately.It gives an ₁-solution of AX+YB=C under additional assumptions.Some numerical examples are given.     

Factored forms for solutions of AX − XB = C and X − AXB = C in companion matrices

The main concern of this paper is linear matrix equations with block-companion matrix coefficients. It is shown that general matrix equations AX ? XB = C and X ? AXB = C can be transformed to equations whose coefficients are block companion matrices: $\text{C}\text{?}{}_{\mathrm{L}}\text{X?XC}{}_{\mathrm{M}}\text{=}\text{diag}\text{[I 0…0]}$ and $\text{X?}\text{C}\text{?}{}_{\mathrm{L}}\text{XC}{}_{\mathrm{M}}\text{=}\text{diag}\text{[I 0…0]}$, respectively, where ?_L and C_M stand for the first and second block-companion matrices of some monic r × r matrix polynomials L(λ) = λ^sI + Σ^s?1_j=0λ^jL_j and M(λ) = λ^tI + Σ^t7minus;1_j=0λ^jM_j. The solution of the equat with block companion coefficients is reduced to solving vector equations Sx = ?, where the matrix S is r²l × r²l[l = max(s, t)] and enjoys some symmetry properties.     

Roth's theorems over commutative rings

In 1952, W.E. Roth showed that matrix equations of the forms AX?YB = C and AX?XB = C over fields can be solved if and only if certain block matrices built from A, B, and C are equivalent or similar. We show here that these criteria remain valid over arbitrary commutative rings. To do this, we use standard commutative algebra methods to reduce to the case of Artinian rings, where a simple argument with     

An index classification of M-matrices

An M-matrix as defined by Ostrowski [5] is a matrix that can be split into A = sI ? B, where s > 0, B ? 0, with s ? r(B), the spectral radius of B. Following Plemmons [6], we develop a classification of all M-matrices. We consider v, the index of zero for A, i.e., the smallest nonnegative integer n such that the null spaces of Aⁿ and Aⁿ⁺¹ coincide. We characterize this index in terms of convergence properties of powers of s^?1B. We develop additional characterizations in terms of nonnegativity of the Drazin inverse of A on the range of A^v, extending (as conjectured by Poole and Boullion [7]) the well-known property that A^?1?0 whenever A is nonsingular.     

Existence of a low rank or ℋ︁‐matrix approximant to the solution of a Sylvester equation

We consider the Sylvester equation AX?XB+C=0 where the matrix C∈?^n×m is of low rank and the spectra of A∈?^n×n and B∈?^m×m are separated by a line. We prove that the singular values of the solution X decay exponentially, that means for any ε∈(0,1) there exists a matrix X? of rank k=O(log(1/ε)) such that ∥X?X?∥₂?ε∥X∥₂. As a generalization we prove that if A,B,C are hierarchical matrices then the solution X can be approximated by the hierarchical matrix format described in Hackbusch (Computing 2000; 62 : 89–108). The blockwise rank of the approximation is again proportional to log(1/ε). Copyright © 2004 John Wiley & Sons, Ltd.     

The matrix equations AX = C, XB = D

For the pair of matrix equations AX = C, XB = D this paper gives common solutions of minimum possible rank and also other feasible specified ranks.     

On a characterization of irreducibility of a non-negative matrix

Let A denote an n×n matrix with all its elements real and non-negative, and let r_i be the sum of the elements in the ith row of A, i=1,…,n. Let B=A?D(r₁,…,r_n), where D(r₁,…,r_n) is the diagonal matrix with r_i at the position (i,i). Then it is proved that A is irreducible if and only if rank B=n?1 and the null space of B^T contains a vector d whose entries are all non-null.     

Minimum product set sizes in nonabelian groups of order pq

Let G be a nonabelian group of order pq, where p and q are distinct odd primes. We analyze the minimum product set cardinality μG(r,s)=min|AB|, where A and B range over all subsets of G of cardinalities r and s, respectively. In this paper, we completely determine μG(r,s) in the case where G has order 3p and conjecture that this result can be extended to all nonabelian groups of order pq. We also prove that for every nonabelian group of order pq there exist 1?r,s?pq such that μG(r,s)>μZ/pqZ(r,s).     

Optimally small sumsets in finite abelian groups

Let G be a finite abelian group of order g. We determine, for all 1?r,s?g, the minimal size μ_G(r,s)=min|A+B| of sumsets A+B, where A and B range over all subsets of G of cardinality r and s, respectively. We do so by explicit construction. Our formula for μ_G(r,s) shows that this function only depends on the cardinality of G, not on its specific group structure. Earlier results on μ_G are recalled in the Introduction.     

Nonsingular solutions of TA−BT=C

When A, B and C are given square matrices and C is of rank one, sufficient conditions are given for every solution to be nonsingular when solutions exist. When C has arbitrary rank, some sufficient conditions are given; and when, additionally, A and B have disjoint spectra, necessary conditions are given.     

Enumeration of matrices of given rank with submatrices of given rank

The authors determine the number of (n+m)×t matrices A¹ of rank r+v, over a finite field GF(q), whose last m rows are those of a given matrix A of rank r+v over GF(q) and whose first n rows have a given rank u.     

On the invertibility of a nearly singular matrix

Let z be a complex variable and let A and B be constant n × n matrices with complex elements. It is shown that A + zB is invertible for all z in a deleted neighborhood of zero if and only if there exist constant n × n matrices such that XA + YB = I and AX + BY = I. A related result is the alternate necessary and sufficient condition that there exist constant X, Y such that XA + YB = I, YAXB = XBYA = 0 and YA is nilpotent.     

r-Numbers completion problems of partial upper canonical form I

A pair (A, B), where A is an n × n matrix and B is an n × m matrix, is said to have the nonnegative integers sequence {r_j}_j=1^p as the r-numbers sequence if r₁ = rank(B) and r_j = rank[B AB … A^j−1 B] − rank[B AB … A^j−2B], 2 ≤ j ≤ p. Given a partial upper triangular matrix A of size n × n in upper canonical form and an n × m matrix B, we develop an algorithm that obtains a completion A_c of A, such that the pair (A^c, B) has an r-numbers sequence prescribed under some restrictions.     

A comment on “minimization of functions of a positive simidefinite matrix A subject to AX = 0”

A common problem in multivariate analysis is that of minimizing a scalar function φ of a positive semidefinite matrix A subject possibly to AX = 0. In this paper it is suggested to replace A by B′B, where B is allowed to vary freely, subject possibly to BX = 0.     

Singular M-matrices and inverse positivity

For a matrix decomposable as A=sI?B, where B?0, it is well known that A^?1?0 if and only if the spectral radius ρ(B)>s. An extension of this result to the singular case ρ(B)=s is made by replacing A^?1 by [A+t(I?AA^D)]^?1, where A^D is the Drazin generalized inverse.     

The decomposition of Cr(K) into the direct sum of subalgebras

Let A and B be closed subalgebras of C_r(X) whose direct sum is C_r(X). Some consequences of this relation are explored in this paper. For example if 1 ?A (as may be assumed) it is shown that the norm of the projection onto A is an odd integer and there is a retraction of X onto the set of common zeros of elements of B.