arXiv subjects
Publications and source records attributed to Juan-Miguel Gracia.
Let $A$ be a square complex matrix and $z$ a complex number. The distance, with respect to the spectral norm, from $A$ to the set of matrices which have $z$ as an eigenvalue is less than or equal to the distance from $z$ to the spectrum of $A$. If these two distances are equal for a sufficiently large finite set of numbers $z$ which are not in the spectrum of $A$, then the matrix $A$ is normal.
In this document I recapitulate some results by Hiriart-Urruty and Ye (1995) concerning the properties of differentiability and the existence of lateral directional derivatives of the multiple eigenvalues of a complex Hermitian matrix function of several real variables, where the eigenvalues are supposed in a decreasing order. Another version of these results was obtained by Ji-guang Sun (1988). This 2020 version has been written following a remark raised by Miloud Sadkane about the presence or not of the value 1/2 in the formulas that give the lateral directional derivatives of multiple singular values. Therefore, I have added Theorem 9. I have also written in more detail the relationship of the reviewed results with those in the bibliography. Finally, I have also put a result of mine (Corollary 11 in [1]) (joint with Armentia and Velasco) which interprets a Lippert's Theorem (2005).
The structured pseudospectra of a matrix A are sets of complex numbers that are eigenvalues of matrices X which are near to A and have the same entries as A at a fixed set of places. The sum of multiplicities of the eigenvalues of X inside each connected component of the structured pseudospectra of A does not depend on X. This fact is known, but not so much as it should be. For this reason, we give here an elementary and detailed proof of the result.