Short Normal Paths and Spectral Variation
Rajendra Bhatia, John Holbrook · Proceedings of the American Mathematical Society · 1985
We introduce the notion of a "short normal path" between matrices $S$ and $T$, that is, a continuous path from $S$ to $T$ consisting of normal matrices and having the same length as the straight line path from $S$ to $T$. By this means we prove that for certain normal matrices $S$ and $T$ the eigenvalues of $S$ and $T$ may be paired in such a way that the maximum distance (in the complex plane) between the pairs is no more than the operator norm $\left \| {S - T} \right \|$. In particular, we generalize and provide a new approach to a recent result of Bhatia and Davis treating the case of unitary $S$ and $T$.