# Is there an orthogonal matrix Q so that the $Q ^ tAQ = D$ and $Q ^ tBQ$ matrices are diagonal

Let $$A = \left(\begin{array}{cc}1&-2 \\ -2&5 \end{array}\right) \in M_n(\mathbb{C})$$

Let $$B = \left(\begin{array}{cc}-3&6 \\ 6&-10 \end{array}\right) \in M_n(\mathbb{C})$$

Is there an orthogonal matrix Q so that the $$Q ^ tAQ$$ and $$Q ^ tBQ$$ matrices are diagonal

• Simultaneously diagonalisable matrices always commute, but your $A$ and $B$ do not. Commented Jun 30, 2019 at 15:41
• Hello @user1932595, what do you think about my solution? Notice that all what you need to do is to do one multiplication and to check whether the product is symmetrical ( ..and you have exact theoretical explanation) Commented Jul 1, 2019 at 7:54

Assume $$Q ^ tAQ=D_1$$ and $$Q ^ tBQ=D_2$$.
Then $$Q ^ tAQQ ^ tBQ=Q ^ tABQ=D_1 D_2$$
what means that $$AB=QD_1D_2Q^t$$ should be symmetric.
You can check that $$AB$$ in this case is not symmetric.
• For diagonal matrices $D \ \$ it holds $\ \ \ \ QDQ^t=(QDQ^t)^t$ Commented Jun 30, 2019 at 17:31
No, since no eigenvector of $$A$$ is also an eigenvector of $$B$$.