Ranks of Solutions of the Matrix Equation AXB = C

The purpose of this article is to solve two problems related to solutions of a consistent complex matrix equation AXB = C : (I) the maximal and minimal ranks of solution to AXB = C , and (II) the maximal and minimal ranks of two real matrices X₀ and X₁ in solution X = X₀ + iX₁ to AXB = C . As applications, the maximal and minimal ranks of two real matrices C and D in generalized inverse (A + iB)^- = C + iD of a complex matrix A + iB are also examined.     

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.     

Sums and products of matrices with prescribed similarity classes

Let F be a field and let A and n × n matrices over F. We study some properties of A^' + B^' and A^'B^', when A^' and B^' run over the sets of the matrices similar to A and B, respectively.     

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.     

Rank equalities related to outer inverses of matrices and applications

A matrix X is called an outer inverse for a matrix A if XAX=X. In this paper, we present some basic rank equalities for difference and sum of outer inverses of a matrix, and apply them to characterize various equalities related to outer inverses, Moore-Penrose inverses, group inverses, Drazin inverses and weighted Moore-Penrose inverses of matrices.     

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.     

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.     

Equality of higher numerical ranges of matrices and a conjecture of Kippenhahn on Hermitian pencils

Let M_n be the algebra of all n × n complex matrices. For 1 k n, the kth numerical range of A M_n is defined by W_k(A) = (1/k)∑_j^k=1x_j^*Ax_j : x₁, …, x_k is an orthonormal set in ⁿ]. It is known that tr A/n = W_n(A) W_n−1(A) W₁(A). We study the condition on A under which W_m(A) = W_k(A) for some given 1 m < k n. It turns out that this study is closely related to a conjecture of Kippenhahn on Hermitian pencils. A new class of counterexamples to the conjecture is constructed, based on the theory of the numerical range.     

Two inequalities for the sum of elements of a matrix

The inequalities su A^t su A^m ≥ su A^p su A^q and A^t+ su A^m≥ su A^p+ su A^q are studied and generalized. Here su A denotes the sum of elements of the square matrix A.     

Doubly stochastic matrices whose powers eventually stop

In this note we characterize doubly stochastic matrices A whose powers A,A²,A³,… eventually stop, i.e., A^p=A^p+1= for some positive integer p. The characterization enables us to determine the set of all such matrices.     

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.     

Height bases, level bases, and the equality of the height and the level characteristics of an M-matrix

Let A be an M-matrix. We introduce the concepts of height basis, level basis, and height-level basis for the generalized nullspace of A. We explore the properties of such bases and of induced matrices. We use these results to prove some new conditions for the equality of the (spectral) height (Weyr) characteristic and the (graph theoretic) level characteristic of A, and to simplify proofs of known conditions. We also prove the existence of a Jordan basis for the generalized nullspace with all chains of maximal length nonnegative.     

The solvability of two linear matrix equations

The solvability conditions of the following two linear matrix equations (i)A₁X₁B₁+A₂X₂B₂+A₃X₃B₃=C,(ii) A₁XB₁=C₁A₂XB₂=C₂ are established using ranks and generalized inverses of matrices. In addition, the duality of the three types of matrix equations

(iii) A₁X₁B₁+A₂X₂B₂+A₃X₃B₃+A₄X₄B₄=C, (iv) A₁XB₁=C₁A₂XB₂=C₂A₃XB₃=C₃A₄XB₄=C₄, (v) AXB+CXD=E are also considered.     

The robertson-taussky inequality revisited

Let a positive definite Hermitian matrix HεMⁿ(C) be decomposed as H=A + iB, with A, B ε M_nm(R). We give two new proofs of the inequality det H ≤ det A (with equality iff B = 0. each of which vields something futher. One exhibits majorization between the eigenvalues of A and H the other allows proof of the permanental analog per H≥per A.     

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.     

Linear operators strongly preserving r-cyclic matrices over semirings

We say that A is an r-cyclic matrix if A^r=I. We investigate the structure of linear operators on matrices over antinegative semirings that map r-cyclic matrices to r-cyclic matrices and non r-cyclic matrices to non r-cyclic matrices.     

Positive diagonal solutions to the Lyapunov equations

We study various stability type conditions on a matrix A related to the consistency of the Lyapunov equation AD+DA^t positive definite, where D is a positive diagonal matrix. Such problems arise in mathematical economics, in the study of time-invariant continuous-time systems and in the study of predator-prey systems. Using a theorem of the alternative, a characterization is given for all A satisfying the above equation. In addition, some necessary conditions for consistency and some related ideas are discussed. Finally, a method for constructing a solution D to the equation is given for matrices A satisfying certain conditions.     

Nonnegative mth roots of nonnegative 0-symmetric idempotent matrices

Nonnegative mth roots of nonnegative 0-symmetric idempotent matrices have been characterized. As an application, a characterization of nonnegative matrices A whose Moore-Penrose generalized inverse A² is some power of A is obtained, thus yielding some well-known theorems.     

Decomposably non-negative matrices

Let A be an mn- by - mn symmetric matrix. Partition A into m²n - by - n blocks and suppose that each of these blocks is also symmetric. Suppose that for every decomposable (rank one) tensor ν ⊗ w, we have (ν ⊗ w)^t A(ν otimes; w) ≥ 0. Here, ν is a column m-tuple and w is a column n-tuple. We study the maximum number of negative eigenvalues such a matrix can have, as well as obtaining alternative characterizations of such matrices.