An Algorithm for Finding a 2-Similarity Transformation from a Numerical Contraction to a Contraction
Daeshik Choi, Anne Greenbaum · SIAM Journal on Matrix Analysis and Applications · 2015
It was shown in [T. Ando, Acta Sci. Mat. (Szeged), 34, pp. 11--15] that any matrix $A$ with numerical radius at most 1 is similar to a contraction (a matrix $T$ with spectral norm at most $1$) via a similarity transformation with condition number at most 2; that is, $A = S T S^{-1}$, where $\| T \| \leq 1$ and $\kappa (S) \equiv \| S \| \cdot \| S^{-1} \| \leq 2$. However, no explicit algorithm was given for producing such a similarity transformation; in this paper, we give a method for constructing such similarity transformations. As a side benefit, the algorithm indicates whether the numerical radius of $A$ is greater than 1 (or greater than some given number $r_0$) and so can be used to determine (sometimes very quickly) whether the numerical radius is greater than a given value.