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
editBased 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.