TITLE: Inverse of a symmetric positive definite matrix
QUESTION [12 upvotes]: If a matrix is symmetric and positive definite, determine if it is invertible and if its inverse matrix is symmetric and positive definite.
I know that "if a matrix is symmetric and positive definite, then its inverse matrix is also positive definite", based on a theorem. But I am not sure how to prove that the matrix even is invertible or that its inverse matrix is also symmetric.
It would really help if someone explained this a bit. Thanks

REPLY [2 votes]: If $Q$ is psd, $$y^T Q y ≥ 0 \quad ∀ y \qquad \& \qquad I = Q^{-1}Q$$because all eigenvalues are positive.
Let $x = Q^{-1}y$ for any y,
$$(Q^{-1}y)^T Q Q^{-1}y \geq 0 ⇒ y^T (Q^{-1})^T y \geq 0$$
and we know psd matrices are symmetric.