# Engineering Analysis/Quadratic Forms

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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[edit]

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.