Engineering Analysis/Quadratic Forms

If we have an n × 1 vector x, and an n × n symmetric matrix M, we can write: 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: if for all x, then the matrix is positive definite. if for all x, then the matrix is positive semi-definite. if for all x, then the matrix is negative definite. if for all x, then the matrix is negative semi-definite. These classifications are used commonly in control engineering.