Normal matrices and an extension of Malyshev's formula
For a given complex matrix \(A\) of order \(n \geq 3\), the \(2\)-norm distance of \(A\) from the set \({\mathcal M}\) of matrices with a zero eigenvalue of multiplicity \(\geq 3\) is investigated. It is proved that for a normal matrix \(A\) this distance is equal to \[ \max_{\gamma_1, \gamma_2 \geq 0, \gamma_3 \in {\mathbf C}} \sigma_{3n-2}(Q(\lambda)), \] where \(Q(\lambda)\) is a \(3n \times 3n\) matrix associated with \(A\) in the following way: \[ Q(\lambda)= \begin{pmatrix} A & \gamma_1 I_n & \gamma_3 I_n \\ 0 & A & \gamma_2 I_n \\ 0 & 0 & A \end{pmatrix}. \] Here, \(\gamma_1, \gamma_2, \gamma_3\) are scalar parameters in \(\gamma = (\gamma_1, \gamma_2, \gamma_3)\) and \(\sigma_i\), \(1 \leq i \leq 3n\), are singular values of \(Q(\lambda)\) in the decreasing order. This result represents an extension of \textit{A. N. Malyshev}'s formula [Numer. Math. 83, 443--454 (1999; Zbl 0972.15011)] for the \(2\)-norm distance of an arbitrary \(n \times n\) matrix from the set \({\mathcal M}\) for the case of normal matrices.
scientific article; zbMATH DE number 2121405