# Complex Eigenvalues/vectors

1. Nov 26, 2009

### Iconate

1. The problem statement, all variables and given/known data
Find the eigenvalues and eigenvectors of A. (Both eigenvalues and eigenvectors are now allowed
to be complex.) Is it diagonalizable? Explain why or why not. If it is diagonalizable, explicitly
find matrices P and D such that
A = PDP−1
where D is a diagonal 2 × 2 matrix.

A = [ 0 -i | i 0 ]

3. The attempt at a solution

I determined that A cannot be diagonalized because, by the characteristic polynomial equation we get $$\lambda$$2 + 1 = 0

Therefore $$\lambda$$1 = -i $$\lambda$$2 = i

plugging $$\lambda$$2 into my matrix A I get:

ix + iy = 0
-ix + iy = 0

but the only solution to this is x=y=0, I get the same result for $$\lambda$$1

Is this correct? I have a feeling this trivial solution is wrong
I tried row reduction, but I still get the same result.

2. Nov 26, 2009

### tiny-tim

so many minuses!

Hi Iconate!

(have a lambda: λ )
Noooo!

3. Nov 26, 2009

### Iconate

Re: so many minuses!

Ahhh I see
my determinant should be
λ2 - (-i)(i) = 0
λ2 + (i2) = 0
λ2 - 1 = 0

thus λ1 = 1 λ1 = -1

Thanks >.<