On the Distance to the Closest Matrix with Triple Zero Eigenvalue

Kh. D. Ikramov ; A. M. Nazari

Matematičeskie zametki, Tome 73 (2003) no. 4, pp. 545-555

Résumé

The 2-norm distance from a matrix $A$ to the set $\mathscr M$ of $(n\times n)$ matrices with a zero eigenvalue of multiplicity $\ge3$ is estimated. If $$ Q(\gamma_1,\gamma_2,\gamma_3)=\begin{pmatrix} A\gamma_1I_n\gamma_3I_n \\0\gamma_2I_n \\00 \end{pmatrix}, \qquad n\ge3, $$ then $$ \rho_2(A,\mathscr M) \ge\max_{\gamma_1,\gamma_2\ge0,\,\gamma_3\in\mathbb C} \sigma_{3n-2}(Q(\gamma_1,\gamma_2,\gamma_3)), $$ where $\sigma_i(\cdot)$ is the $i$th singular value of the corresponding matrix in the decreasing order of singular values. Moreover, if the maximum on the right-hand side is attained at the point $\gamma^*=(\gamma^*_1,\gamma^*_2,\gamma^*_3)$, where $\gamma^*_1\gamma^*_2\ne0$, then, in fact, one has the exact equality $$ \rho_2(A,\mathscr M) =\sigma_{3n-2}(Q(\gamma^*_1,\gamma^*_2,\gamma^*_3)). $$ This result can be regarded as an extension of Malyshev's formula, which gives the 2-norm distance from $A$ to the set of matrices with a multiple zero eigenvalue.