# Thread: Eigen value of skew symmetric matrix

1. ## 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)
AX+AX = 0
AX+kX=0
operating X
XAX+XkX=0
(AX)X+XkX = 0
(kX)X+XkX=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.

2. ## Re: Eigen value of skew symmetric matrix

Here $\displaystyle k$ is just a constant so

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

$\displaystyle 2k|X|^2 = 0$ 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.