Geodesic
Justification of a Malyshev-Type Formula in the Nonnormal Case
Matematičeskie zametki, Tome 78 (2005) no. 2, pp. 241-250
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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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Kh. D. Ikramov; A. M. Nazari. Justification of a Malyshev-Type Formula in the Nonnormal Case. Matematičeskie zametki, Tome 78 (2005) no. 2, pp. 241-250. http://geodesic.mathdoc.fr/item/MZM_2005_78_2_a8/

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