Nonnormal eigenvalues do not bound a quadratic quotient
= Nonnormal eigenvalues do not bound a quadratic quotient
The real <matrix> $B_M=\begin{pmatrix}1&M\\0&1\end{pmatrix}$ has both <eigenvalues> equal to one, whereas $x^TB_Mx/(x^Tx)=1+M/2$ for $x=(1,1)^T$. Hence no universal constant bounds this quotient by the maximum <eigenvalue> of every real <matrix> with real <eigenvalues>.