TABLE OF CONTENTS A ? R m n is said to be a nonnegative matrix whenever each a ij ? 0, and this is denoted by writing A ? 0 . In general, A ? B means that each a ij ? b ij . Similarly, A is a positive matrix when each a ij > 0 , and this is denoted by writing A > 0 . More generally, A > B means that each a ij > b ij .
Applications abound with nonnegative and positive matrices. In fact, many of the applications considered in this text involve nonnegative matrices. For example, the connectivity matrix C in Example 3.5.2 (p. 100) is nonnegative. The discrete Laplacian L from Example 7.6.2 (p. 563) leads to a nonnegative matrix because (4 I ? L) ? 0 . The matrix e A t that defines the solution of the system of differential equations in the mixing problem of Example 7.9.7 (p. 610) is nonnegative for all t ? 0 . And the system of difference equations p( k) = Ap( k ? 1) resulting from the shell game of Example 7.10.8 (p. 635) has a nonnegative coefficient matrix A .
Since nonnegative matrices are pervasive, it s natural to investigate their properties, and that s the purpose of this chapter. A primary issue concerns the extent to which the properties A
