Definition: A symmetric (Hermitian) matrix $A$ is positive-definite if $x^T Ax > 0$ (or $x^*Ax > 0$) for all nonzero vectors $x$. Equivalently, all eigenvalues are positive. Positive-definite matrices are always invertible and have a unique Cholesky decomposition.
**Definition:** A symmetric (Hermitian) matrix $A$ is **positive-definite** if $x^T Ax > 0$ (or $x^*Ax > 0$) for all nonzero vectors $x$. Equivalently, all eigenvalues are positive. Positive-definite matrices are always invertible and have a unique Cholesky decomposition.