Abstract:
Let $A$ be a complex matrix of order $n$, $n\ge3$. We associate with $A$ the $3n$ $$
Q(\gamma)=\begin{pmatrix}
A&\gamma_1I_n&\gamma_3I_n
\\
0&A&\gamma_2I_n
\\
0&0&A
\end{pmatrix},
$$
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 decreasing order. Under certain assumptions on $A$, the authors have proved earlier that the 2-norm distance from $A$ to the set of matrices with a zero eigenvalue of multiplicity at least 3 is equal to max
$$
\max_{\gamma_1,\gamma_2,\gamma_3\in\mathbb C}\sigma_{3n-2}(Q(\gamma)).
$$
Now, the justification of this formula for the distance is given for an arbitrary matrix $A$.
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