If a matrix of the form I + B is singular, then ||B|| ≥ 1 for every subordinate norm. I need some guidance showing that: If a matrix of the form I+B is singular, where I is the identity matrix, then for any subordina...--prophetes.ai

If a matrix of the form I + B is singular, then ||B|| ≥ 1 for every subordinate norm. I need some guidance showing that: If a matrix of the form I+B is singular, where I is the identity matrix, then for any subordinate norm $\|\cdot\|$, $\|B\|\geq1$.

If the matrix is singular, then for some $v$ with $\|v\| = 1$, we have $(I+B)v = 0$, that is, $Bv = -v$. Hence $\|B\| = \sup_{\|y\|\le 1} \|By\| \ge \|Bv\| = \|-v\|=1$.