Matrix Eigenvalues

Computing $det(A-I\lambda) = 0$ (where $A$ is the 3x3 matrix above) yields the following characteristic equation:

$(1-\lambda)((1-\lambda)^2 - 0) - (-1)(0 - 1) + (0)(0 + 1) = 0;$

or $(1-\lambda)^3 - 1 = 0;$

or $-3\lambda + 3\lambda^{2} - \lambda^{3} = 0$ ;

or $\lambda(-3 + 3\lambda - \lambda^{2}) = 0;$

or $\lambda = 0, \frac{3 \pm \sqrt{3}i}{2}$ .

Hence: $\frac{|\lambda_{1} + \lambda_{2} + \lambda_{3}|}{|\lambda_{1}| + |\lambda_{2}| + |\lambda_{3}|} = \frac{|0 + 2(3/2)|}{|0| + 2|\sqrt{(3/2)^2 + (\sqrt{3}/2)^2}|} = \frac{|3|}{2|\sqrt{3}|} = \boxed{\frac{\sqrt{3}}{2}}.$

The sum of the Eigenvalues is the Trace of the matrix, given by adding the diagonals from top left to bottom right. Then, after finding the bottom part of the ratio (adding the absolute value of the complex Eigenvalues), you can find the ratio.