I tried to prove the following statement:
$A$ a negative definite matrix, then $A^{-1}$ exists and $A^{-1}$ is negative definite as well.
Proof:
Let $A \in M_n(\mathbb{K})$ be a negative definite matrix $\Rightarrow$ All eigenvalues of $A : \lambda_{1 \le i\le n} < 0$. Especially $\lambda_i \neq 0$. $\Rightarrow$ $det(A) \neq 0 \Rightarrow A$ is invertible.
Then $A^{-1}$ has eigenvalues $\lambda_i^{-1} \Rightarrow \lambda_i^{-1} <0 \Rightarrow$ negative definite matrix.
Is that correct?
