When is the largest singular value is strictly greater than all eigenvalues?

Question

I found a proof of the fact that the largest singular value is always greater or equal to $|\lambda|_{\max}$.

In what cases equality holds and in what cases it is strictly greater?

Thank you.

Answer 1

Equality holds for symmetric matrices (it is a consequence of the spectral theorem).

Example of strict inequality: consider the matrix $$\left[\begin{array}{}0&1\\0&0\end{array}\right].$$ Its singular values are $0$, $1$, but all eigenvalues are zero.

There are some non symmetric matrices whose singular values coincide with the largest (in absolute value) eigenvalue though. For instance $$\left[\begin{array}{}0&1\\-1&0\end{array}\right]$$ (the singular values are all $1$, and the eigenvalues are $\pm i$).

I don’t know a general criterion for when $\sigma_1$ and $|\lambda|_{\max}$ coincide.