Chapter 7.3, Problem 7P

Show that

${({\sigma }_{1}X+{\sigma }_{2}Y)}^2-4({\sigma }_1{\sigma }_2-{\alpha }_1{\alpha }_2)XY={({\sigma }_{1}X-{\sigma }_{2}Y)}^2+4{\alpha }_1{\alpha }_{2}XY.$

Hence conclude that the eigenvalues given by Eq. (38) can never be complex.

${\lambda }_{1,2}=\frac{-({\sigma }_{1}X+{\sigma }_{2}Y)\pm \sqrt{{({\sigma }_{1}X+{\sigma }_{2}Y)}^2-4({\sigma }_1{\sigma }_2-{\alpha }_1{\alpha }_2)XY}}{2}$ (38)