I'm wondering what the difference between canonical forms of a matrix are; in particular why they aren't all equivalent to diagonal matrices. For example, we know that all matrices have an upper triangular form. If we use elementary row and column operations, then wouldn't the matrix become diagonal? Not sure how this changes diagonalisability, since the pre and post-multiplication matrices are still both square and invertible.
Similarly for other canonical forms such as Jordan Normal Form. Under this logic, why aren't all matrices diagonalisable? Do I have the definition of diagonalisability wrong?