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Given a diagonalizable positive matrix A A , there exists a positive matrix with the same spectrum as A A , and with arbitrarily prescribed elementary divisors, provided that elementary divisors corresponding to nonreal eigenvalues occur in conjugate pairs. It is also shown that a similar result holds for doubly stochastic matrices.

Sufficient conditions are given for stability and nonnegativity of solutions of a system of differential equations, in particular, of comparison differential equations. Finally, it has been shown that the comparison differential equations represent the mathematical models for competitive processes in biological, physical and social sciences.

We relate some minimax functions of matrices to some spectral functions of matrices. If A A is a nonnegative n × n n \times n matrix, υ ( A ) \upsilon (A) is the game-theoretic value of A A , and ρ ( A ) \rho (A) is the spectral radius of A A , then υ ( A ) ≤ ρ ( A ) \upsilon (A) \leq \rho (A) . Necessary and sufficient conditions for υ ( A ) = ρ ( A ) \upsilon (A) = \rho (A) are given. It follows that if A A is nonnegative and irreducible and n > 1 n > 1 , then υ ( A ) > ρ ( A ) \upsilon (A) > \rho (A) . Also, if, for a real matrix A A and a positive matrix B B , υ ( A , B ) = sup X inf Y X T A Y / X T B Y \upsilon (A,B) = {\sup _X}{\inf _Y}{X^T}AY/{X^T}BY over probability vectors X X and Y Y , then for nonnegative, nonsingular A A and positive B B , ρ ( A B ) = [ υ ( A − 1 , B ) ] − 1 \rho (AB) = {[\upsilon ({A^{ - 1}},B)]^{ - 1}} .

Nonnegative rectangular matrices having nonnegative W W -weighted group inverses are characterized. Our techniques suggest an interesting approach to extend the earlier known results on λ \lambda -monotone square matrices to rectangular ones. We also answer a question of characterizing nonnegative matrices having a nonnegative solution X X where (1) A = A X A A = AXA , (2) X = X A X X = XAX , (3) ( A X ) (AX) is 0 0 -symmetric, (4) ( X A ) (XA) is 0 0 -symmetric. In particular, we obtain theorems of Berman-Plemmons and Plemmons-Cline characterizing nonnegative matrices A A with a nonnegative Moore-Penrose inverse. Matrices having nonnegative generalized inverses are of interest in the study of finding nonnegative best approximate solutions of linear systems. Such matrices are of considerable interest in statistics, numerical linear algebra and mathematical economics.

A decomposition of nonnegative matrices with nonnegative group inverses has been obtained. This decomposition provides a new approach to the solution of problems relating to nonnegative matrices with nonnegative group inverses. As a consequence, a number of results are derived. Our results, among other things, answer a question of Berman, extend the theorems of Berman and Plemmons, DeMarr and Flor.

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.

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