I'm trying to use Mathematica to show that the eigenvalues of $U$ are $\pm\dfrac{1-i}{\sqrt{2}} $, where
$U = (I + T + iS)(I - T- iS)^{-1}$ where $ S = \left( \begin{matrix} 1 & 1 \\ 1 & 1 \\ \end{matrix} \right), T = \left( \begin{matrix} 0 & 1 \\ -1 & 0 \\ \end{matrix} \right)$.
Here is my input but Mathematica gives the eigenvalues as $-1,-1$. What went wrong? Thank you. I spent the past half hour looking over my input but can't find the problem.
S = {{1, 1}, {1, 1}}
T = {{0, 1}, {-1, 0}}
(I + T + I*S).Inverse[I - T - I*S]
(*
Output {{-1 - 2 I, 2 I}, {-2 I, -1 + 2 I}}
*)
Eigenvalues[{{-1 - 2 I, 2 I}, {-2 I, -1 + 2 I}}]
(*
Output: {-1, -1}
*)
Iwhich is defined as the built-in symbol for the imaginary unit. I bet that in your problemIstands for some other matrix, maybe the unit matrix. Then there is also an i in your formula, which I assume really is the imaginary unit. But you have to clean up your notation first. – Jens Jan 5 at 18:52Iyou've used is the square root of -1. What you need isIdentityMatrix[2]. Look atEigenvalues@(IdentityMatrix@2 + T + I*S).Inverse[ IdentityMatrix@2 - T - I*S] // ComplexExpand– rm -rf♦ Jan 5 at 18:54