Why Must A Matrix be Symmetric for Orthogonal Diagonalization

Question

So far, all we are doing in class is determine if the matrix A is symmetric, find the basis for the eigenspace P, and apply Gram Schmidt for it to be orthogonal. My question is; why must A be symmetric in the first place?

In the end of the day, we will still apply Gram Schmidt to solve for orthogonal matrices - wont that mean that the inverse of P is the transpose? Does that imply A wont produce a diagonal matrix if it wasn't symmetric? Any idea why?

Thanks

Answer 1

You should not think that any matrix can be diagonalized. The fact that a symmetric matrix $A$ can be diagonalized, with an orthogonal diagonalizing matrix $P$ (i.e., $P^{-1}=P^T$), is the content of the so-called "Spectral Theorem" for symmetric matrices.

If you allow $A$ to be non-symmetric then you may find out that $A$ does not admit orthogonal diagonalizing matrices, or even that it cannot be diagonalized altogether.