I have a matrix that is the canonical [1,0][0,1] -- rotated 30 degrees, and scaled with [2,0][0,.5].
I am curious to know why the eigenvectors of this matrix lose their orthogonality. My understanding is that the eigenvalues are what that bring the determinant to 0, and that the eigenvectors, when multiplied by the original matrix, are only scaled versions. So wherein this process does orthogonality get lost?
I am curious about this in the contest of eigendecompositions. Given $A = PDP^{-1}$, I see that AD is a scaled version of A. But I don't understand why plotting P alone reveals vectors with a new angle. I suspect symmetry of vectors has something to do with all this, but I'm not sure.
Thank you in advance.