## Deriving QR and Shur decomposition

I have a problem. Given

$\ Ax = \lambda x$

and the QR factorization

$\ x = Qr = Q \begin{bmatrix} r_1 \\ 0 \end{bmatrix}$

Using these two equations I want to SHOW that

$Q^H AQ = \begin{bmatrix} \lambda & T_{12} \\ 0 & T_{22} \end{bmatrix}$

Now Q is unitary i.e
$\ Q^H Q = QQ^H = I$

Can you show me how to do this?