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 $$ Q(\gamma)=\begin{pmatrix} A&\gamma_1I_n&\gamma_3I_n \\0&A&\gamma_2I_n \\0&0&A \end{pmatrix}, $$ 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.

Bibliographie

[1] Malyshev A. N., “A formula for the $2$-norm distance from a matrix to the set of matrices with multiple eigenvalues”, Numer. Math., 83 (1999), 443–454 | DOI | MR | Zbl

[2] Ikramov Kh. D., Nazari A. M., “O rasstoyanii do blizhaishei matritsy s troinym sobstvennym znacheniem nul”, Matem. zametki, 73:4 (2003), 545–555 | MR | Zbl