1
$\begingroup$

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!

$\endgroup$
  • $\begingroup$ 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 $\endgroup$ – Batman Jun 20 '15 at 2:41
1
$\begingroup$

If your Hermitian matrix has all real entries (is symmetric) then here are a few applications:

  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 $\Sigma=(X-\mu)^T(X-\mu)$.

  3. The graph Laplacian matrix is diagonally dominant and thus PSD.

  4. Positive semidefiniteness defines a partial order on the set of symmetric matrices (this is the foundation of semidefinite programming).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.