A matrix A epsilon LRn is skewsymmetric if AT A Show that a

A matrix A epsilon L(R^n) is skew-symmetric if A^T = -A. Show that all eigenvalues of a skew-symmetric matrix are imaginary.

Solution

Consider any matrix A as a matrix over the set of complex numbers.

Then we have that for all x,yCⁿ

x,yCⁿ,

{Ax, y } ={x, A*y}

where {Ax,y{ represents the standard inner product and A* represents the adjoint of A,

BUt adjoint of A = Transpose of A (since A is real)

{Ax,y} = {x, A^Ty} = {x, -Ay}

If x is the eigen vector corresponding to c,

set y =x

{Ax,x} = {cx,x} = c||x||²

Also we have

-{x, Ax} = -{x,cx} = -c bar {x, x} = -c bar ||x||²

These two must be equal as c is the eigen value, and x is the eigen vector.

So we get ||x|| =0 Or c = -c bar or c =0

Which proves that c can be only 0 or purely imaginary.