Spectral norm of matrices with complex eigenvalues

Question

Suppose that $M$ is a square, invertible matrix with eigenvalues $\lambda_1, \ldots, \lambda_n$ where the lambda's can possibly be complex. Suppose that $\lambda_{\max}(M)$ is complex valued. How is the spectral norm going be defined in this case?

Answer 1

Despite its name, the spectral norm of a matrix has nothing to do with the spectrum of the matrix. It is the largest singular value of the matrix. Unless the matrix is normal, you really cannot infer the largest singular value from the largest-sized eigenvalue.

E.g. consider $M=\pmatrix{\lambda&m\\ 0&\lambda}$ for a fixed $\lambda$. As $\lambda$ is fixed, the spectrum of $M$ is fixed, but $\|M\|\to\infty$ when $|m|\to\infty$.