Engineering Analysis/Quadratic Forms

If we have an n &times; 1 vector x, and an n &times; 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.