Page 97 - Matrices theory and applications

5
Nonnegative Matrices

In this chapter matrices have real entries in general. In a few specified cases,
entries might be complex.

5.1 Nonnegative Vectors and Matrices

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Definition 5.1.1 Avector x ∈ IR is nonnegative, and we write x ≥ 0,
if its coordinates are nonnegative. It is positive, and we write x> 0,if its
coordinates are (strictly) positive. Furthermore, a matrix A ∈ M n×m(IR)
(not necessarily square) is nonnegative (respectively positive) if its entries
are nonnegative (respectively positive); we again write A ≥ 0 (respectively
A> 0). More generally, we define an order relationship x ≤ y whose
meaning is y − x ≥ 0.
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Definition 5.1.2 Given x ∈ CC ,welet |x| denote the nonnegative vector
whose coordinates are the numbers |x j |. Similarly, if A ∈ M n (CC),the
matrix |A| has entries |a ij |.
Observe that given a matrix and a vector (or two matrices), the triangle
inequality implies

|Ax|≤ |A|·|x|.
Proposition 5.1.1 A matrix is nonnegative if and only if x ≥ 0 implies
Ax ≥ 0. It is positive if and only if x ≥ 0 and x =0 imply Ax > 0.

Proof