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# 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.