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Mat. Zametki, 2005, Volume 78, Issue 2, Pages 241–250 (Mi mz2579)  

This article is cited in 2 scientific papers (total in 2 papers)

Justification of a Malyshev-Type Formula in the Nonnormal Case

Kh. D. Ikramov, A. M. Nazari

M. V. Lomonosov Moscow State University

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 $A$.

DOI: https://doi.org/10.4213/mzm2579

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English version:
Mathematical Notes, 2005, 78:2, 219–227

Bibliographic databases:

UDC: 519.6
Received: 26.12.2003
Revised: 08.12.2004

Citation: Kh. D. Ikramov, A. M. Nazari, “Justification of a Malyshev-Type Formula in the Nonnormal Case”, Mat. Zametki, 78:2 (2005), 241–250; Math. Notes, 78:2 (2005), 219–227

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. Kressner D. Mengi E. Nakic I. Truhar N., “Generalized Eigenvalue Problems With Specified Eigenvalues”, IMA J. Numer. Anal., 34:2 (2014), 480–501  crossref  mathscinet  zmath  isi  scopus  scopus
    2. Karow M. Mengi E., “Matrix Polynomials With Specified Eigenvalues”, Linear Alg. Appl., 466 (2015), 457–482  crossref  mathscinet  zmath  isi  scopus  scopus
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