On the Distance to the Closest Matrix with Triple Zero Eigenvalue

The 2-norm distance from a matrix A to the set ${\mathcal{M}}$ of n × n matrices with a zero eigenvalue of multiplicity ≥3 is estimated. If $$Q(\gamma _1 ,\gamma _2 ,\gamma _3 ) = \left( {\begin{array}{*{20}c} A & {\gamma _1 I_n } & {\gamma _3 I_n } \\ 0 & A & {\gamma _2 I_n } \\ 0 & 0 & A \\ \end{array} } \right), n \geqslant 3,$$ then $$\rho _2 (A,{\mathcal{M}}) \geqslant {\mathop {max}\limits_{\gamma _1 ,\gamma _2 \geqslant 0,\gamma _3 \in {\mathbb{C}}}} \sigma _{3n - 2} (Q(\gamma _1 ,\gamma _2 ,\gamma _3 )),$$ where σⁱ(·)is the ith 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 $\gamma _1^ * \gamma _2^ * \ne 0$ , then, in fact, one has the exact equality $$\rho _2 (A,{\mathcal{M}}) = \sigma _{3n - 2} (Q(\gamma _1^ * ,\gamma _2^ * ,\gamma _3^ * )).$$ 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.