Normal Matrices and an Extension of Malyshev's Formula

Abstract

Let A be a complex matrix of order n with n ≥ 3. We associate with A the 3n × 3n matrix $Q\left( {\gamma } \right) = \left( \begin{gathered} A \gamma _1 I_n \gamma _3 I_n \hfill \\ 0 A \gamma _2 I_n \hfill \\ 0 0 A \hfill \\ \end{gathered} \right)$ where $\gamma _1 ,\gamma _2 ,\gamma _3$ are scalar parameters and γ=(γ₁,γ₂,γ₃). Let σ_i, 1 ≤ i ≤ 3n, be the singular values of Q(γ) in the decreasing order. We prove that, for a normal matrix A, its 2-norm distance from the set $\mathcal{M}$ of matrices with a zero eigenvalue of multiplicity at least 3 is equal to $\mathop {max}\limits_{\gamma _1 ,\gamma _2 \geqslant 0,\gamma _3 \in \mathbb{C}} \sigma _{3n - 2} (Q\left( \gamma \right)).$ This fact is a refinement (for normal matrices) of Malyshev's formula for the 2-norm distance from an arbitrary n × n matrix A to the set of n × n matrices with a multiple zero eigenvalue.