Quadratic Forms & Positive Definiteness

Quadratic Form is also called "Weighted Length", given by the following equation:

$$x^TAx = \sum_{ij}x_ix_jA_{ij} = scalar$$

​Positive Definite (P.D.) matrix has all eigen values greater than 0. A positive definite matrix has the property, $x^TAx > 0, \forall x \neq 0$​. For example, consider the following:

$$A = I = \begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix}$$

​Thus, $x^TAx = x^Tx > 0, \forall x \neq 0$​.

A positive semi definite (P.S.D.) matrix has all eigen values greater than or equal to 0. A positive semi definite matrix has the property that $x^TAx \geq 0, \forall x \neq 0$​. For example, consider the following:

$$A = \begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix}, \hspace{0.1cm} x = \begin{bmatrix} 0\\ 0\\ 1 \end{bmatrix}$$

​We can compute to observe $x^TAx = 0$​.

Similar definitions exist for negative definite and negative semi definite matrices as well.