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## Homework Statement

X= 1st row: (0, 1, 0, 0), 2nd row: (1, 0, 0, 0), 3rd row: (0, 0, 0, 1-i), 4th row: (0, 0, 1+i, 0)

Find the eigenvalues and

**eigenvectors**of the matrix X.

## Homework Equations

|X-λI|=0 (characteristic equation)

(λ is the eigenvalues, I is the identity matrix)

(X-λI)V=0 (V is the corresponding eigenvector V= (V1, V2, V3, V4))

## The Attempt at a Solution

Apologies firstly for my poor attempt at a matrix...

I have found the eigenvalues by solving the characteristic equation to be

λ1= +1,

λ2=-1

λ3= +√2

λ4=-√2

and then using the second equation above to find the eigenvectors of X.

The problem I'm having is that when trying to find the eigenvectors, I end up with 4 equations, 2 of which have components V1, V2 only, and the other 2 equations have components V3 and V4 only so there's no way to eliminate them and find the normalised eigenvectors. The subsequent parts of the question ask to find the exponential of X and so without the eigenvectors I cannot achieve this. I've only ever been taught how to do this for a 2x2 which is relatively simple but that procedure doesn't seem to be working for me here.

Thanks in advance!