I have two $n \times n$ matrices A and B. They have same eigenvectors but not necessarily same eigenvalues and both of them are diagonalizable. I want to show that for any real numbers c and d, $E=cA+dB$ is also diagonalizable.
My question is:
Even though they have same eigenvectors, does not the order in which the eigenvectors are put determine what diagonalizing matrix they have? Or in other words, do they need to have same diagonalizing matrix? How do I approach this problem?