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

${\displaystyle x^{T}Mx=a}$

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 ${\displaystyle x^{T}Mx>0}$ for all x, then the matrix is positive definite.
2. if ${\displaystyle x^{T}Mx\geq 0}$ for all x, then the matrix is positive semi-definite.
3. if ${\displaystyle x^{T}Mx<0}$ for all x, then the matrix is negative definite.
4. if ${\displaystyle x^{T}Mx\leq 0}$ for all x, then the matrix is negative semi-definite.

These classifications are used commonly in control engineering.