[Math] proof of positive semi-definiteness of the precision matrix (inverse of the covariance matrix)

covariancelinear algebramatrices

I would like to know how to prove that the inverse of a covariance matrix $\Sigma^{-1}=\Omega$ is positive semi-definite too.

My second question can we prove that for any matrix $A\in\cal{M}_{nm}$ we have the determinant $|A^T~\Omega~A|\ge0$.

Best Answer

Hint. Every covariance matrix is positive semidefinite. Hence every invertible covariance matrix is positive definite. Now, every positive definite matrix is unitarily diagonalisable. Therefore ...

For your second question, presumably the matrices are real. Consider the quantity $x^TA^T\Omega Ax$. Do you see that it is always nonnegative?

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