What makes a matrix positive definite

Give the definition of a positive-definite matrix and state its main properties.

A symmetric $A$ with $x^\top Ax>0$ for every nonzero $x$. Equivalent tests: all eigenvalues are positive; all leading principal minors are positive (Sylvester); $A=LL^\top$ with $L$ lower triangular and positive diagonal (Cholesky). Consequences: $A$ is invertible with $A^{-1}$ also positive definite, $\det A>0$, diagonal entries are positive, and sums of positive-definite matrices stay positive definite.