Positive Definite Matrix Question

Question

I am having some little difficulties to show the following:

Let $M$ be a symmetric positive square matrix. I must show that $I-M$ is positive definite matrix if and only if $M^{-1}-I$ is positive definite.

Answer 1

Note that M is Symmetric Positive Definite, Implying that the eigenvalues are all positive. This argument works backwards too. (This would not be true if M were not Symmetric in general) \begin{align} \rightarrow\\ \because M \text{ is PD, }\\ \lambda_i&>0\quad \forall i\\ eig(I-M)&=(1-\lambda_i)\\\because(I-M)\text{ is PD, }\\ (1-\lambda_i)&>0\\ \implies 1&>\lambda_i>0\quad \forall i\\ eig(M^{-1}-I)&=\dfrac{1}{\lambda_i}-1\\ &>0\\ \implies (M^{-1}-I) \text{ is PD} \end{align} Similarly, you can do the reverse direction.