I'm struggling with a problem from Boas's Mathematical Methods in the Physical Sciences. The question is, for a 2×2 matrix M s.t. M is real, not symmetric, with eigenvalues real and not equal, show that the eigenvectors of M are not orthogonal.
I've tried to manipulate MC = CD in Einstein notation (C, a matrix that diagonalizes M and D, the matrix of eigenvalues) with little luck. I've also tried manipulation of arbitrary elements of M*C and C*D to get an expression for the inner product of the eigenvectors, but I'm not finding any relationships that force the dot product to be non-zero when the off-diagonal elements of M aren't equal.
Any suggestions for a better approach would be appreciated. I've already turned in the assignment, but this question's still bugging me!
Best Answer
By contradiction if there's two orthogonal eigenvectors then there's a change matrix $C$ which's an orthogonal matrix i.e. $C^{-1}=C^t$ and then $$M=CDC^t$$ is a symmetric matrix. Contradiction.