Can the closest non singular matrix be seen as the closest positive/negative definite matrix?

Question

Now I understand that positive definite matrices are non-singular as the determinant is non-0, could the closest non singular matrix be treated as the closest positive definite matrix? I think I am basically asking are all non singular matrices positive definite, actually they can be negative definite as well I guess so could it be reworded as the closest positive or negative definite matrix?

Answer 1

Consider the matrix $A = \begin{bmatrix}1&0&0\\0&0&0\\0&0&-1\end{bmatrix}$. Clearly, $A$ is singular.

For any $\epsilon \neq 0$, the matrix $A(\epsilon) = \begin{bmatrix}1&0&0\\0&\epsilon&0\\0&0&-1\end{bmatrix}$ is invertible and $\|A(\epsilon)-A\| = \epsilon$. So there is no closest non-singular matrix to $A$, but we can find non-singular matrices that are arbitrarily close to $A$.

However, for any positive definite matrix $X$, we must have $X_{3,3} > 0$, and thus $\|X-A\| \ge |X_{3,3}-A_{3,3}| = X_{3,3}+1 > 1$. Similarly, for any negative definite matrix $Y$, we must have $Y_{1,1} < 0$, and thus $\|Y-A\| \ge |Y_{1,1}-A_{1,1}| = 1-Y_{1,1} > 1$.

Thus, there are no positive definite or negative definite matrices within a distance of $1$ of $A$.

So "the closest non-singular matrix" is not the same thing as "the closest positive definite or negative definite matrix".