On the distance to the closest matrix with triple zero eigenvalue (Q1886129)

The 2-norm distance from a complex \(n\times n\) matrix \(A\) to the set \({\mathcal M}\) of \(n\times n\) matrices with a zero eigenvalue of multiplicity \(\geqslant 3\) is estimated. If \(Q=\left(\begin{smallmatrix} A & \gamma_1 I_n & \gamma_3 I_n \\ 0 & A & \gamma_2 I_n \\ 0 & 0 & A\end{smallmatrix}\right) \), \(n\geqslant 3\), then \(\rho_2(A,{\mathcal M})\geqslant \max \{ \sigma _{3n-2} (Q(\gamma_1,\gamma_2,\gamma_3)) \mid \gamma_1,\gamma_2\geqslant 0, \;\gamma_3\in {\mathbb C}\}\), where \(\sigma_i(\cdot)\) is the \(i\)th singular value. Moreover, if the maximum is attained at the point \(\gamma^*=(\gamma_1^*, \gamma_2^*,\gamma_3^*)\) and \(\gamma_1^*\gamma_2^* \neq 0\), then \(\rho_2(A,{\mathcal M})=\sigma _{3n-2} (Q(\gamma_1,\gamma_2,\gamma_3))\). This result partially generalizes one result of \textit{A. N. Malyshev} [Numer. Math. 83, 443--454 (1999; Zbl 0972.15011)] where he obtained a 2-distance from \(A\) to the set of matrices with a multiple zero eigenvalue.