Inverse of symmetric matrices

Question

I know that not all symmetric matrices are invertible, but under which conditions are they invertible ? and do all symmetric matrices have to be either positive or negative definite in order to have inverse ??

Thanks in advance

Answer 1

If you talk about real symmetric matrices then they are invertible iff all eigenvalues are different from zero. (They don't need to be all all positive or all negative). The inverse of $CDC^{-1}$ is simply $C D^{-1} C^{-1}$ where $D$ is diagonal.

Answer 2

The invertibility depends upon the rank, that is the n-by-n matrix is invertible $\iff$ $rank=n$.

That's true for any kind of matrix and of course also for symmetric matrix.

Also note that a symmetric matrix is invertible $\iff$ if all eigenvalues are $\neq 0$.