Engineering Analysis/Quadratic Forms

If we have an n × 1 vector x, and an n × n symmetric matrix M, we can write:

x^TMx = a

Where a is a scalar value. Equations of this form are called quadratic forms.

Matrix Definiteness

Based on the quadratic forms of a matrix, we can create a certain number of categories for special types of matrices:

1. if x^TMx > 0 for all x, then the matrix is positive definite.
2. if $x^TMx \ge 0$ for all x, then the matrix is positive semi-definite.
3. if x^TMx < 0 for all x, then the matrix is negative definite.
4. if $x^TMx \le 0$ for all x, then the matrix is negative semi-definite.

These classifications are used commonly in control engineering.