This page covers how to tell if a matrix is positive definite, what it means for it to be positive definite, and the geometric view of it

Positive Definite Matrices

Given a $2\times 2$ symmetric matrix

$$ A = \begin{bmatrix} a & b\\ b & c \end{bmatrix} $$

there are four ways to tell if its positive definite:

1. Eigenvalue test: $\lambda_1 > 0, \lambda_2 > 0$
2. Determinants test: $det(A) > 0$
3. Pivot test: $a > 0, \frac{ac - b^2}{a} > 0$
4. $x^TAx >0$ for every column $x \neq 0$ in $\R^2$