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This chapter discusses the semigroups of nonnegative matrices. The matrix multiplication is associative, and the product of two non-negative matrices is again a nonnegative matrix. The most important applications of the material in the chapter involve the solvability of certain nonnegative matrix equations arising in the areas of mathematical economics and mathematical programming. A subgroup G of a semigroup T is called a maximal subgroup of T if it is not properly contained in any other subgroup of T . This chapter discusses the algebraic theory of semigroups of nonnegative matrices. Idempotent elements play a fundamental role in algebraic semigroup theory. Every matrix in the semigroup N n of nonnegative matrices is not regular. The tool to be used to characterize Green's relations for regular elements in N n would be that of rank. However, this tool is more of a vector space notion and is too sophiscated to characterize Green's relations on the entire semigroup N n .

The literature on dynamic systems in economics is vast, and an important part of that deals with systems of differential or of difference equations where the Jacobian of the right-hand side is a Metzler matrix, i. e., a matrix whose off-diagonal elements are non-negative. Such matrices have a wide range of applicability in dynamic economic models, in input-output analysis, in stability analysis of systems of excess demands governing price changes, and in multi-sector and multi-national Keynesian income determination models. Oskar Morgenstern early perceived the importance of such models and encouraged research in them, as seen in the papers by Y.K. Wong and M.A. Woodbury in Morqenstern (3). For a later survey, see (2).

Homogeneous 2D state space models whose variables are always nonnegative are described by a pair of nonnegative square matrices (A,B). In the paper, we discuss some spectral and combinatorial properties under particular assumptions on the structure of the matrix pair, like finite memory, separability and property L. >

Three sufficient conditions for the irreducibility of a matrix A are given, which for nonnegative A are also necessary.

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

Liao Xiao Xin (1987) gave necessary and sufficient conditions for the stability of a class of interval matrices. It can easily be shown that these results follow directly from a theorem relating stable Metzler matrices and quasidominant negative diagonal matrices by McKenzie (1966) which states (see also Siljak 1978) A Metzler matrix A is stable if and only if it is quasidominant negative diagonal. As the author considers Metzler matrices, Theorems 1,2 and 3 are obvious. Also, a theorem by Fiedler and Ptak (1962) concerning a pair of Minkowski matrices (see also Siljak 1978) leads to the same results.

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