- #1
mindarson
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Homework Statement
I'm trying to find the eigenvalues/eigenvectors of the unitary matrix
A = (1/√5){{1,2},{2i,-i}}
Homework Equations
det[A-λI]=0
AA* = I (where A* denotes the adjoint of A)
The Attempt at a Solution
I have tried to do this straightforwardly, as I would any other matrix, by using the equation above. However, I end up with the following quadratic equation:
λ^2+λi-λ-5i = 0
My first thought was to group this according to real and imaginary parts and then set both to zero:
λ^2 - λ + (λ-5)i = 0
But this actually gives me 3 eigenvalues and none of them is correct, anyway. So that's a dead end (although I don't understand why; the logic makes sense to me: if a complex number is zero, then it's real part must be zero and it's imaginary part must be zero. Where did I go wrong here? Maybe in not accounting for the possibility of complex eigenvalues, so my grouping is not correct?)
My next tactic was to put it into the classic form of a quadratic equation, like so:
λ^2+(i-1)λ-5i=0
and then apply the good old quadratic equation with a = 1, b = (i-1), and c = -5i. This gives me
(1/2)(-i+1±3*Sqrt(2i))
This is not only not the correct answer, but I don't even know how to make sense of it. (What is the square root of i? Just the 1/4th power of -1, I suppose?) Can anyone offer some guidance as to where I've gone wrong here?
Am I perhaps supposed to be using the fact that the matrix is unitary?