Equality Conditions for the Singular Values of $3\times 3$ Matrices
Matematičeskie zametki, Tome 80 (2006) no. 2, pp. 187-192
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Let $\Gamma_a$ be an upper triangular $3\times 3$ matrix with diagonal entries equal to a complex scalar $a$. Necessary and sufficient conditions are found for two of the singular values of $\Gamma_a$ to be equal. These conditions are much simpler than the equality $\operatorname{discr}\varphi=\nobreak 0$, where the expression in the left-hand side is the discriminant of the characteristic polynomial $\varphi$ of the matrix $G_a=\Gamma_a^*\Gamma_a$. Understanding the behavior of singular values of $\Gamma_a$ is important in the problem of finding a matrix with a triple zero eigenvalue that is closest to a given normal matrix $A$.
Keywords:
upper triangular matrix, singular value of a matrix, spectral distance, characteristic equation.
Mots-clés : normal matrix
Mots-clés : normal matrix
@article{MZM_2006_80_2_a3,
author = {Kh. D. Ikramov},
title = {Equality {Conditions} for the {Singular} {Values} of $3\times 3$ {Matrices}},
journal = {Matemati\v{c}eskie zametki},
pages = {187--192},
year = {2006},
volume = {80},
number = {2},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/MZM_2006_80_2_a3/}
}
Kh. D. Ikramov. Equality Conditions for the Singular Values of $3\times 3$ Matrices. Matematičeskie zametki, Tome 80 (2006) no. 2, pp. 187-192. http://geodesic.mathdoc.fr/item/MZM_2006_80_2_a3/