# What are some Applications of Hermitian Positive Definite matrices?

I am learning about Hermitian positive definite matrices and the way that they can be decomposed with Cholesky decomposition. I have learned that these matrices deserve special attention for how often they are used when dealing with numerical linear algebra.

My question is simply: what (real-world) applications are there that use Hermitian positive definite matrices? For example, in statistics the covariance matrix of a multi-variate probability distribution is positive semi-definite. I am also inclined to think that positive definite matrices occur in finite-difference methods of solving PDEs. What other applications of such matrices are there? All thoughts are welcome and examples/sources are greatly appreciated!

• Pretty much everything comes down to a positive definite or positive semi-definite matrix operations at some point in numerical linear algebra. Check any numerical linear algebra book for detials – Batman Jun 20 '15 at 2:41

1. If the Hessian of $f$ is PSD (positive semidefinite) then $f$ is convex.
2. The covariance matrix is always PSD since it's formed as $\Sigma=(X-\mu)^T(X-\mu)$.