ar X iv : 0 80 2 . 38 74 v 2 [ m at h . FA ] 2 S ep 2 00 8 On low rank perturbations of complex matrices and some discrete metric spaces .

The article is devoted to different aspects of the question: ”What can be done with a complex-valued matrix by a low rank perturbation?” From the works of Thompson [15] we know how the Jordan normal form can be changed by a rank k perturbation, see Theorem 2. Particulary, it follows that one can do everything with a geometrically simple spectrum by a rank 1 perturbation, see Corollary 1. But the situation is quite different if one restricts oneself to normal matrices, see Theorem 3 and Corollary 2. We think that Corollary 2 may be considered as a finite dimension analogue of the continuous spectrum conservation under compact perturbations in Hilbert spaces. For unitary and self-adjoint matrices the inequality of Corollary 2 is the only restrictions on ”what can be done with a spectrum by a rank k perturbation”, see Theorem 4. We don’t know if there is an analogue of Theorem 4 for normal matrices. It is worth to mention that Corollary 2 for self-adjoint matrices follows from Cauchy interlacing theorem [2]. Theorem 4 is related with the converse Cauchy interlacing theorem [6]. The spectrum of H1 + H2 with known spectra of self-adjoint matrices H1 and H2 is studied a lot, see [9] and the bibliography therein. Although the complete set of restrictions on the spectrum H1+H2 known in this situation, we are not sure that there is an easy proof of Theorem 4 using results of [9]. Although Theorem 2 should be known (see, for example, [13], where Theorem 2 formulated in one direction), we will give a proof here, manly because our proof falls in a general framework , which is also used in the proof of Theorem 4. Let us describe the framework. The set Cn×n of all complex n×nmatrices (set of self-adjoint matrices) we equip with the arithmetic distance, d(A,B) = rank(A−B) (see [3]). The arithmetic distance is geodesic for these cases. The spectral properties of matrices, such as Weyr characteristics and spectra (multiset) also may be considered as a metric spaces with distance, related to the arithmetic distance on matrices, see Section 2. These distances also turn out to be geodesic. Then we prove Theorem 2 (Theorem 4) for rank(A− B) = 1 and the general results will follow from Proposition 1.