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

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 $A$.