I have a problem. Given

$\displaystyle \ Ax = \lambda x $

and the QR factorization

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

Using these two equations I want to SHOW that

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

Now Q is unitary i.e

$\displaystyle \ Q^H Q = QQ^H = I $

Can you show me how to do this?