Linear Algebra with Applications, 10th Edition

6.7 Positive Definite Matrices

In Section 6.6, we saw that a symmetric matrix is positive definite if and only if its eigenvalues are all positive. These types of matrices occur in a wide variety of applications. They frequently arise in the numerical solution of boundary value problems by finite difference methods or by finite element methods. Because of their importance in applied mathematics, we devote this section to studying their properties.

Recall that a symmetric $n\times n$ matrix A is positive definite if ${\mathbf{x}}^{T}A\mathbf{x}>0$ for all nonzero vectors x in ${ℝ}^n$. In Theorem 6.6.2, symmetric positive definite matrices were characterized by the condition that all their eigenvalues are positive. This characterization can be used to establish the following properties: ...