I've two small questions I'm quite stuck on, and I'd appreciate your help.
1) Any real square matrix, all whose eigenvalues are real, having an orthonormal basis of eigenvectors, is symmetric.
Now I know that due to The Spectral Theorem, if a matrix is symmetric then we can find an orthonormal basis of its eigenvectors. However, this is like trying to prove A -> B only knowing that B -> A, which isn't much good. I've tried to find a contradiction, but to no avail. I'm inclined to say that it is true.
2) If A is a symmetric matrix, and F is the matrix whose columns are an orthonormal basis of eigenvectors of A, then F is symmetric.
Now for this, I tried using the inner product but didn't get anywhere. I know of orthogonal matrices that aren't symmetric (like the Sin theta Cos theta one), but I don't think that's sufficient, as who's to say that matrices with that property always contain the orthonormal basis of eigenvectors of some matrix A.