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 Zh. Vychisl. Mat. Mat. Fiz., 2005, Volume 45, Number 3, Pages 383–390 (Mi zvmmf679)

On the singular values of a special 3-by-3 matrix: sufficient conditions for monotonicity along a ray

Kh. D. Ikramov

Faculty of Computational Mathematics and Cybernetics, Moscow State University, Moscow, 119992, Russia

Abstract: Let $\Gamma_a$ be a 3-by-3 upper triangular matrix with all the diagonal entries equal to $a$. For a fixed $a$, the singular values of $\Gamma_a$ are examined as functions of the off-diagonal entries $\gamma_{ij}$ ($i<j$). It is shown that at most three stationary points ($t=0$ not included) are possible for all the singular values of $\Gamma_a$ combined on the ray $R(\alpha,\beta,\mu)$: $\gamma_{12}=\alpha t$, $\gamma_{23}=\beta t$, $\gamma_{13}=\mu t$, $t\ge 0$. Sufficient conditions are obtained for the monotonicity of all the singular values or for the monotonicity of only the extremal ones along the ray $R(\alpha,\beta,\mu)$. The understanding of the behavior of the singular values of $\Gamma_a$ is important in the problem of finding a matrix with a triple zero eigenvalue closest to a given normal matrix $A$.

Key words: spectral norm, normal matrix, singular values, stationary point, discriminant.

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English version:
Computational Mathematics and Mathematical Physics, 2005, 45:3, 366–373

Bibliographic databases:
UDC: 519.614

Citation: Kh. D. Ikramov, “On the singular values of a special 3-by-3 matrix: sufficient conditions for monotonicity along a ray”, Zh. Vychisl. Mat. Mat. Fiz., 45:3 (2005), 383–390; Comput. Math. Math. Phys., 45:3 (2005), 366–373

Citation in format AMSBIB
\Bibitem{Ikr05} \by Kh.~D.~Ikramov \paper On the singular values of a special 3-by-3 matrix: sufficient conditions for monotonicity along a~ray \jour Zh. Vychisl. Mat. Mat. Fiz. \yr 2005 \vol 45 \issue 3 \pages 383--390 \mathnet{http://mi.mathnet.ru/zvmmf679} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2161479} \zmath{https://zbmath.org/?q=an:1082.15019} \transl \jour Comput. Math. Math. Phys. \yr 2005 \vol 45 \issue 3 \pages 366--373