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It is shown that if a linear transformation T on the space of n × n complex matrices maps nonnegative matrices into nonnegative matrices and preserves the spectrum of each nonnegative matrix, then T ( A ) = P −1 AP or T ( A ) = P −1 A T P for all matrices A and a fixed nonnegative generalized permutation matrix P .

Let λ be any subset of {1,2, 3, 4} containing 1 and consider the matrix equations (1) and AXA=A, (2) XAX=X, (3) AX=(AX) T (4) XA=(XA) T where A is an m×n real matrix. By a λ-inverse of A is meant a solution of the equations (i)ieλ. A result on binary relations is given and is used to obtain a characterization of all nonnegative matrices A having a nonnegative λ-inverse, for all possible λ.

Summary The problem of finding all nonnegative definite solutions to the “Riccati” algebraic equation PA + A'P - PBB'P + C'C = 0 is solved in general. The class of such solutions is shown to constitute a distributive lattice. Finally an algorithm to compute those solutions without checking all general solutions is proposed.

A "measure of irreducibility" of a nonnegative matrix is introduced and applied to obtaining an estimate from below for the distance of the Perron eigenvalue from any other eigenvalue of such a matrix.

This paper contains an attempt to develop for discrete semigroups of infinite order matrices with nonnegative elements a simple theory analogous to the Perron-Frobenius theory of finite matrices. It is assumed throughout that the matrix is irreducible, but some consideration is given to the periodic case. The main topics considered are (i) nonnegative solutions to the inequalities ^xj (r>0)

Matrices for which the group inverse exists and is nonnegative are studied. Such matrices are characterized in terms of a generalization of monotonicity. In particular, nonnegative matrices with this property are characterized in terms of their nonnegative rank factorizations.

Results concerning generalized inverses of n × n matrices over the Boolean algebra of order two are applied to the study of directed graphs and nonnegative matrices. Maximal systems of directed graphs are discussed. Also, necessary conditions are obtained for a nonnegative matrix to have a nonnegative generalized inverse.

Let A m × n , B m × n , X n × 1 {A_{m \times n}},{B_{m \times n}},{X_{n \times 1}} , and Y m × 1 {Y_{m \times 1}} be matrices whose entries are nonnegative real numbers and suppose that no row of A and no column of B consists entirely of zeroes. Define the operators U, T and T’ by ( U X ) i = X i − 1 [ or ( U Y ) i = Y i − 1 ] {(UX)_i} = X_i^{ - 1}[{\text {or}}{(UY)_i} = Y_i^{ - 1}] , T = U B t U A T = U{B^t}UA and T ′ = U A U B t T’ = UAU{B^t} . T is called irreducible if for no nonempty proper subset S of { 1 , ⋯ , n } \{ 1, \cdots ,n\} it is true that X i = 0 , i ∈ S ; X i ≠ 0 , i ∉ S {X_i} = 0,i \in S;{X_i} \ne 0,i \notin S , implies ( T X ) i = 0 , i ∈ S ; ( T X ) i ≠ 0 , i ∉ S {(TX)_i} = 0,i \in S;{(TX)_i} \ne 0,i \notin S . M. V. Menon proved the following Theorem. If T is irreducible, there exist row-stochastic matrices A 1 {A_1} and A 2 {A_2} , a positive number θ \theta , and two diagonal matrices D and E with positive main diagonal entries such that D A E = A 1 DAE = {A_1} and θ D B E = A 2 t \theta DBE = A_2^t . Since an analogous theorem holds for T’, it is natural to ask if it is possible that T’ be irreducible if T is not. It is the intent of this paper to show that T’ is irreducible if and only if T is irreducible.

We generalize a result stating that a nonnegative finite square matrix has a nonnegative inverse if and only if it is the product of a permutation matrix by a diagonal matrix. We consider column-finite infinite matrices and give a simple proof using elementary ideas from the theory of partially ordered linear algebras. In [1] the authors show that a nonnegative square matrix has a nonnegative inverse if and only if its entries are all zero except for a single positive entry in each row and column. In this note we generalize this result and simplify the proof as well. Let A denote the real linear algebra of all column-finite infinite matrices with real entries. We partially order A as follows: [aij] 10 and note that 1 +d_ (1 +d)2=yxyx

Necessary and sufficient conditions are given in order that a nonnegative matrix have a nonnegative MoorePenrose generalized inverse.

Suppose à is a nonnegative invertible matrix with a positive diagonal Diag ( A ¯ ) > 0 (\bar A) > 0 and y ¯ > 0 \bar y > 0 is a positive vector. Let A = D − 1 A ~ A = {D^{ - 1}}\tilde A and y = D − 1 y ~ y = {D^{ - 1}}\tilde y . If 0 > 2 y − A y 0 > 2y - Ay , then 2 y − A y ≦ x ≦ y 2y - Ay \leqq x \leqq y , where A − 1 y {A^{ - 1}}y .

Suppose A is an n-square matrix over the real numbers such that all principal minors are nonzero. If A is nonnegative, then necessary and sufficient conditions are determined for A to be factored into a product L-U, where L is a lower triangular nonnegative matrix and U is an upper triangular nonnegative matrix with ui, = 1. These conditions are given in terms of the nonnegativity of certain almost-principal minors of A.

A characterization of a class of totally nonnegative matrices whose inverses are M-matrices is given. It is then shown that if A is nonnegative of order n and A-1 is an M-matrix, then the almost principal minors of A of all orders are nonnegative.

Nonnegative matrices characterization by diagonal products to imply properties of stochastic matrices