Geodesic
On the singular values of a special 3-by-3 matrix: sufficient conditions for monotonicity along a ray
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 45 (2005) no. 3, pp. 383-390 
Kh. D. Ikramov. On the singular values of a special 3-by-3 matrix: sufficient conditions for monotonicity along a ray. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 45 (2005) no. 3, pp. 383-390. http://geodesic.mathdoc.fr/item/ZVMMF_2005_45_3_a2/
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     title = {On the singular values of a special 3-by-3 matrix: sufficient conditions for monotonicity along a~ray},
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Voir la notice de l'article provenant de la source Math-Net.Ru

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 $\gamma_{ij}$ ($i). It is shown that at most three stationary points ($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 $A$.

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