Linear Algebra with Applications, 10th Edition

6.8 Nonnegative Matrices

In many of the types of linear systems that occur in applications, the entries of the coefficient matrix represent nonnegative quantities. This section deals with the study of such matrices and some of their properties.

Definition

An $n\times n$ matrix A with real entries is said to be nonnegative if $a_{\mathit{\text{ij}}}\ge 0$ for each i and j and positive if $a_{\mathit{\text{ij}}}>0$ for each i and j.

Similarly, a vector $\mathbf{x}={(x_1,\text{. . . , }{x}_n)}^T$ is said to be nonnegative if each $x_i\ge 0$ and positive if each $x_i>0$.

For an example of one of the applications of nonnegative matrices, we consider the Leontief input–output models.

Application 1 The Open Model

Suppose that there are n industries producing n different products. Each industry requires input of the products from ...