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On the Distance to the Closest Matrix with Triple Zero Eigenvalue
Kh. D. Ikramov,
A. M. Nazari M. V. Lomonosov Moscow State University
Abstract:
The 2-norm distance from a matrix
$A$ to the set
$\mathscr M$ of
$(n\times n)$ matrices with a zero eigenvalue of multiplicity
$\ge3$ is estimated. If
$$
Q(\gamma_1,\gamma_2,\gamma_3)=\begin{pmatrix}
A&\gamma_1I_n&\gamma_3I_n
\\0&A&\gamma_2I_n
\\0&0&A
\end{pmatrix},
\qquad
n\ge3,
$$
then
$$
\rho_2(A,\mathscr M)
\ge\max_{\gamma_1,\gamma_2\ge0,\,\gamma_3\in\mathbb C}
\sigma_{3n-2}(Q(\gamma_1,\gamma_2,\gamma_3)),
$$
where
$\sigma_i(\cdot)$ is the
$i$th 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
$\gamma^*_1\gamma^*_2\ne0$, then, in fact, one has the exact equality
$$
\rho_2(A,\mathscr M)
=\sigma_{3n-2}(Q(\gamma^*_1,\gamma^*_2,\gamma^*_3)).
$$
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.
UDC:
519.6 Received: 20.05.2002
DOI:
10.4213/mzm202
Fulltext:
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