Complex Eigenvalues and Eigenvectors of a 2x2 Matrix: Diagonalizable or Not?

[Iconate]

Homework Statement

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 ]

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.

 

[tiny-tim]

so many minuses!

Hi Iconate!

(have a lambda: λ )

A = [ 0 -i | i 0 ]

… by the characteristic polynomial equation we get $$\lambda$$² + 1 = 0

Noooo!

 

[Iconate]

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

thus λ₁ = 1 λ₁ = -1

Thanks >.<