Justification of a Malyshev-Type Formula in the Nonnormal Case

Kh. D. Ikramov ; A. M. Nazari

Matematičeskie zametki, Tome 78 (2005) no. 2, pp. 241-250

Résumé

Let $A$ be a complex matrix of order $n$, $n\ge3$. We associate with $A$ the $3n$ $$ Q(\gamma)=\begin{pmatrix} A\gamma_1I_n\gamma_3I_n \\ 0\gamma_2I_n \\ 00 \end{pmatrix}, $$ where $\gamma_1,\gamma_2,\gamma_3$ are scalar parameters and $\gamma=(\gamma_1,\gamma_2,\gamma_3)$. Let $\sigma_i$, $1\le i\le3n$, be the singular values of $Q(\gamma)$, in decreasing order. Under certain assumptions on $A$, the authors have proved earlier that the 2-norm distance from $A$ to the set of matrices with a zero eigenvalue of multiplicity at least 3 is equal to max $$ \max_{\gamma_1,\gamma_2,\gamma_3\in\mathbb C}\sigma_{3n-2}(Q(\gamma)). $$ Now, the justification of this formula for the distance is given for an arbitrary matrix $A$.