# Not getting the required eigenvalues [closed]

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}
*)

-
You should never use capital single letters as user-defined symbols in Mathematica - it invariably leads to confusion as in this case where you have I which is defined as the built-in symbol for the imaginary unit. I bet that in your problem I stands 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 18:52
The I you've used is the square root of -1. What you need is IdentityMatrix[2]. Look at Eigenvalues@(IdentityMatrix@2 + T + I*S).Inverse[ IdentityMatrix@2 - T - I*S] // ComplexExpand – rm -rf Jan 5 at 18:54
Voting to close as TL – belisarius Jan 5 at 18:54
If this question hadn't be too localized it would have been a duplicate, e.g. mathematica.stackexchange.com/questions/14726/… – Artes Jan 5 at 19:01

## closed as too localized by belisarius, Jens, Artes, rm -rf♦Jan 5 at 19:02

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