Geodesic
Justification of a Malyshev-Type Formula in the Nonnormal Case
Matematičeskie zametki, Tome 78 (2005) no. 2, pp. 241-250 Cet article a éte moissonné depuis la source Math-Net.Ru

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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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     author = {Kh. D. Ikramov and A. M. Nazari},
     title = {Justification of a {Malyshev-Type} {Formula} in the {Nonnormal} {Case}},
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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/

[1] Malyshev A. N., “A formula for the $2$-norm distance from a matrix to the set of matrices with multiple eigenvalues”, Numer. Math., 83 (1999), 443–454 | DOI | MR | Zbl

[2] Ikramov Kh. D., Nazari A. M., “O rasstoyanii do blizhaishei matritsy s troinym sobstvennym znacheniem nul”, Matem. zametki, 73:4 (2003), 545–555 | MR | Zbl

[3] Ikramov Kh. D., Nazari A. M., “Normalnye matritsy i obobschenie formuly Malysheva”, Matem. zametki, 75:5 (2004), 652–662 | MR | Zbl

[4] Ikramov Kh. D., Nazari A. M., “O blizhaishei matritse s troinym sobstvennym znacheniem nul”, Vestn. MGU. Ser. 15. Vychisl. matem., kibern., 2003, no. 4, 20–24 | Zbl

[5] Kato T., Perturbation theory for linear operators, Springer-Verlag, Berlin, 1966 | MR