Normal Matrices and an Extension of Malyshev''s Formula

Kh. D. Ikramov ; A. M. Nazari

Matematičeskie zametki, Tome 75 (2004) no. 5, pp. 652-662

Résumé

Let $A$ be a complex matrix of order $n$ with $n\ge3$. We associate with $A$ the $(3n\times 3n)$ matrix $$ Q(\gamma)=\begin{pmatrix} A\gamma_1I_n\gamma_3I_n \\0\gamma_2I_n \\00 \end{pmatrix}, $$ 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 the decreasing order. We prove that, for a normal matrix $A$, its 2-norm distance from the set $\mathscr M$ of matrices with a zero eigenvalue of multiplicity at least 3 is equal to $$ \max_{\gamma_1,\gamma_2\ge0,\gamma_3\in\mathbb C} \sigma_{3n-2}(Q(\gamma)). $$ This fact is a refinement (for normal matrices) of Malyshev"s formula for the 2-norm distance from an arbitrary $(n\times n)$ matrix $A$ to the set of $(n\times n)$ matrices with a multiple zero eigenvalue.