It seems that when I multiply a matrix by a rotation (orthogonal) matrix, the eigen values change. This is counter-intuitive to me since multiplying by a rotation matrix is simply expressing the column vectors in a new coordinate system. Also, aren't eigen values supposed to be properties of the vector space, supposedly independent of the chosen basis? Can someone provide an intuitive explanation for why the eigen values change?
EDIT: This is what I meant when I said a rotation matrix expresses a vector it is multiplied by in a new coordinate system:
import numpy as np
theta=np.pi/4
rot=np.array([[np.cos(theta),-np.sin(theta)],[np.sin(theta),np.cos(theta)]])
## Expresses the vector [1,-1] in the coordinate system rotated by 45 degrees
np.dot(rot,np.array([1,-1]))
## Result is [1.414,0]