Here is just a constant so
So your expression gives
and so
Another thing you can do is to use
and just do
So the eigenvalues of A are either zero or they come in pairs.
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.