Geodesic
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 
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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/

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