Equality conditions for the singular values of \(3 \times 3\) matrices with one-point spectrum (Q881078)

The author investigates the singular values of matrices of the form \[ \Gamma_{a}=\left[ \begin{matrix} a & \gamma_{1} & \gamma_{3}\\ 0 & a & \gamma_{2}\\ 0 & 0 & a \end{matrix} \right] . \] It is enough to consider the cases \(a=0,1\) and (after a unitary similarity by a suitable diagonal matrix) we may assume that \(\gamma_{1}\) and \(\gamma_{2}\) are nonnegative reals. Under these conditions the author proves that if \(a=1\) then \(\Gamma_{a}\) has a pair of equal singular values if and only if \(\gamma_{3}\) is also real and \(\gamma_{1}^{2}=\gamma_{2}^{2}=\gamma_{3} ^{2}/(1+\gamma_{3})\). If \(a=0\) then \(\Gamma_{a}\) has a pair of equal singular values if and only if either \(\gamma_{1}\gamma_{2}=0\) or \(\gamma_{3} =\gamma_{1}-\gamma_{2}=0\).