Construct matrix Let $B$ any square matrix. Is possible to construct an invertible matrix $Q_B$ such that
$$\|Q_BBQ_B^{-1}\|_2\ \leq\ \rho(B)?$$
Thanks in advance for the help.
Edit: $Q_B$ only need to be invertible, not orthogonal.

Since $\rho(B)=\rho(Q_BBQ_B^{-1})$, you are essentially asking whether $B$ is similar to some $A$ such that $\|A\|_2=\rho(A)$, Yet, $\|A\|_2=\rho(A)$ if and only if $A$ is the scalar multiple of a unitary matrix. So, if $B$ is not similar to the scalar multiple of a unitary matrix (e.g. when $B$ has at least two eigenvalues of different moduli, or when $B$ is not normal), the construction of $Q_B$ is impossible.
