Abstract:
Let $A$ be a complex matrix of order $n$ with $n\ge3$. We associate with $A$ the $(3n\times 3n)$ matrix
$$
Q(\gamma)=\begin{pmatrix}
A&\gamma_1I_n&\gamma_3I_n
\\0&A&\gamma_2I_n
\\0&0&A
\end{pmatrix},
$$
where $\gamma_1,\gamma_2,\gamma_3$ are scalar parameters and $\gamma=(\gamma_1,\gamma_2,\gamma_3)$. Let $\sigma_i$, $1\le i\le3n$, be the singular values of $Q(\gamma)$ in the decreasing order. We prove that, for a normal matrix $A$, its 2-norm distance from the set $\mathscr M$ of matrices with a zero eigenvalue of multiplicity at least 3 is equal to
$$
\max_{\gamma_1,\gamma_2\ge0,\gamma_3\in\mathbb C}
\sigma_{3n-2}(Q(\gamma)).
$$
This fact is a refinement (for normal matrices) of Malyshev"s formula for the 2-norm distance from an arbitrary $(n\times n)$ matrix $A$ to the set of $(n\times n)$ matrices with a multiple zero eigenvalue.
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