0
$\begingroup$

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

$\endgroup$
2
  • 2
    $\begingroup$ Simultaneously diagonalisable matrices always commute, but your $A$ and $B$ do not. $\endgroup$
    – user1551
    Commented Jun 30, 2019 at 15:41
  • $\begingroup$ 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) $\endgroup$
    – Widawensen
    Commented Jul 1, 2019 at 7:54

2 Answers 2

Reset to default
1
$\begingroup$

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.

$\endgroup$
1
  • $\begingroup$ For diagonal matrices $D \ \ $ it holds $ \ \ \ \ QDQ^t=(QDQ^t)^t$ $\endgroup$
    – Widawensen
    Commented Jun 30, 2019 at 17:31
0
$\begingroup$

No, since no eigenvector of $A$ is also an eigenvector of $B$.

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .