Suppose $A$ is a diagonalizable matrix, then we can find an invertible matrix $T$ such that $A=TDT^{-1}$, where $D$ is a diagonal matrix with entries be the eigenvalues of $A$. For simplicity, we can assume $A$ is non-negative with entries bounded above by $c$. My question is how to find the $T$ such that $\|T\|\|T^{-1}\|$ is minimized, and is it possible to give an upper bound of the minimum value? Here $\|\cdot\|$ is some matrix norm, say the induced $l_2$-norm?
I know each column of $T$ is an eigenvector of $A$, but I have no idea what to do next.