Eigen value of skew symmetric matrix

We were trying to find the eigen value of a skew symmetric matrix:

so we proceeded as :

AX=kX X!=0 for some k as eigen value

and A`=-A

so A`+A = 0

operating by X (matrix)

A`X+AX = 0

A`X+kX=0

operating X`

X`A`X+X`kX=0

(AX)`X+X`kX = 0

(kX)`X+X`kX=0

here we ran into trouble as to the definition of k`.

If we take k`=k

we get that k=0 (i.e. eigen value is 0)

But skew symmetric matrix can have 0 as well as imaginary eigen values, which we were unable to show.

Is there some other way of doing it ?

Thanks.

Re: Eigen value of skew symmetric matrix

Here $\displaystyle k$ is just a constant so

$\displaystyle (kX)' = k X'$

So your expression gives

$\displaystyle 2k|X|^2 = 0$ and so $\displaystyle k = 0$

Another thing you can do is to use

$\displaystyle A' = -A$

and just do

$\displaystyle A'X = -A X = -k X$

So the eigenvalues of A are either zero or they come in $\displaystyle \pm$ pairs.