Let $\Theta$ be a positive-definite matrix, and $\hat\Theta$ be a matrix with the same dimension as $\Theta$. If the spectrum norm of $\Theta - \hat\Theta$ converges to 0 with large probability, why can we say that $\hat\Theta$ is also positive-definite with large probability?
This fact is used in many papers, e.g. this one says Theorem 1 implies that are $\hat\Theta$ and $\hat\Omega$ are positive-definite with large probability. Why does this hold? What is the intuition behind it? Will the same result hold if we replace the spectrum norm with any other norms?
Matrix perturbation theory might explain it. The intuition is that two close matrix also have close eigenvalue.
And since all matrix norms are equivalent up to a constant multiplier, this conclusion should also hold under other norms.