In his Unit 3 Exam Review, Strang gives the theorem that a matrix $A$ has orthogonal eigenvectors if and only if $AA^T=A^TA$, and then lists symmetric, antisymmetric and orthogonal matrices as meeting this condition.
However, I know that in the symmetric case it's the eigenspaces that are guaranteed to be orthogonal, not the eigenvectors.
Is it correct to assume that he meant to say eigenspaces and not eigenvectors?
So -- I have a -- you should say "why?" and I'll at least answer why for case one, maybe case two, the checking the Eigen -- that the eigenvectors are perpendicular, I'll leave to, the -- to the book. But let's just realize what -- well, first I have to say, it -- it could happen, like for the identity matrix -- there's a symmetric matrix. Its eigenvalues are certainly all real, they're all one for the identity matrix. What about the eigenvectors? Well, for the identity, every vector is an eigenvector. So how can I say they're perpendicular? What I really mean is the -- they -- this word are should really be written can be chosen perpendicular. That is, if we have -- it's the usual case. If the eigenvalues are all different, then each eigenvalue has one line of eigenvectors and those lines are perpendicular here. But if an eigenvalue's repeated, then there's a whole plane of eigenvectors and all I'm saying is that in that plain, we can choose perpendicular ones. So that's why it's a can be chosen part, is -- this is in the case of a repeated eigenvalue where there's some real, substantial freedom. But the typical case is different eigenvalues, all real, one dimensional eigenvector space, Eigen spaces, and all perpendicular.