I have two $n \times n$ matrices $A$ and $B$. They have the same eigenvectors but not necessarily the 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?
Best Answer
Let $V$ be a matrix containing the eigenvectors (of which there are $n$, where these are $n \times n$ matrices). Then $$ V^{-1} A V = D_1 $$ is diagonal, as is $V^{-1} B V = D_2$. (It'd be a really good exercise for you to figure out why,) If you now compute, you get $$ V^{-1}(cA + dB)V = (cV^{-1}A + dV^{-1}B) V = c V^{-1}AV + dV^{-1}BV = cD_1 + dD_2 $$ which is again diagonal. $$D_{(i,i)} = cD_{1(i,i)} + dD_{2(i,i)}$$