Sufficient condition for a matrix to be positive definite Is a sufficient condition for a $2\times 2$ matrix
$$\left(\begin{array}{cc}a&b\\b&d\end{array}\right)$$ to be positive definite that $a >0$ and $ad > b^2$ ?

Slightly high-brow route: if a symmetric matrix $\mathbf A$ is positive definite, then the matrix $\mathbf X\mathbf A\mathbf X^\top$ is positive definite as well (Sylvester). Now, consider the decomposition
$$\begin{pmatrix}a&b\\b&d\end{pmatrix}=\begin{pmatrix}1&0\\\frac{b}{a}&1\end{pmatrix}\cdot\begin{pmatrix}a&0\\0&d-\frac{b^2}{a}\end{pmatrix}\cdot\begin{pmatrix}1&0\\\frac{b}{a}&1\end{pmatrix}^\top$$
How does one check if a diagonal matrix is positive definite?
