Matrix , M has all eigenvalues $\in$ [0,1], but on simulation, I can see that the largest singular value is >1.
a) Can anyone give an example of such a matrix in toy cases like 2x2
b) Can some property be pointed out that proves that even though eigenvalues $ \in [0,1]$, but singular values $\notin [0,1]$ i.e. $\exists \sigma > 1$.
In symmetric case, we have $ \sigma_{max} = max\{| \lambda_{max}| ,| \lambda_{min}| \}$, thus we know that the largest singular value in this case would be 1.
Also, in non-symmetric case we know $\sigma_{max} = \lambda_{max}(M^{T}M)^{\frac{1}{2}} $, so when we have $\lambda_{max}(M) >1$, we can get $ \sigma_{max}(M) > \lambda_{max}(M)$.
But how to find relation between $\lambda_{max} > \sigma_{max}$ in case $\lambda_{max}(M) = 1$
$\pmatrix{0&2\\ 0&0}$ gives a simple example. The largest singular value of a matrix is always bounded below by the spectral radius. When they are equal, the matrix is called radial. A complete characterisation of radial matrices is given by