 Nonnormal eigenvalues do not bound a quadratic quotient

The real matrix $B_M=\left(\begin{array}{cc}1& M\\ 0& 1\end{array}\right)$ has both eigenvalues equal to one, whereas $x^T{B}_{M}x/(x^{T}x)=1+M/2$ for $x=(1,1{)}^T$. Hence no universal constant bounds this quotient by the maximum eigenvalue of every real matrix with real eigenvalues.