Completely baffled by this question involving putting matrices in matrices

Question

This is homework, so only hints please.

Let $A\in M_{m\times m}(\mathbb{R})$ , $B\in M_{n\times n}(\mathbb{R})$ . Suppose there exist orthogonal matrices $P$ and $Q$ such that $P^{T}AP$ and $Q^{T}BQ$ are upper triangular. Let $C$ be any $m\times n$ matrix. Then show that there is an orthogonal $R$ , not depending on $C$ , such that $$R^{T} \begin{bmatrix}A & C\\ 0 & B \end{bmatrix}R$$ is upper triangular.

I haven't done any proofs before using matrices containing matrices in them. I have absolutely no idea where to begin. Any ideas?

Answer 1

Take

$$R = \operatorname{diag}(P,Q) = \begin{bmatrix} P \\ & Q \end{bmatrix},$$

and see what happens. Ask if you get stuck.

Answer 2

The elements of a matrix must be add-able and multiply-able (and the addition and multiplication have to satisfy some field properties).

Matrices themselves are add-able and multiply-able (except that AB isn't necessarily BA), so they can be elements of another matrix.

$\begin {bmatrix} A & B \\ C & D \end{bmatrix} \begin {bmatrix} E & F \\ G & H \end{bmatrix} = \begin {bmatrix} A E + B G & A F + B H \\ C E + D F & C G + D H \end{bmatrix}$

In your problem, we want to find

$\begin {bmatrix} X & Y \\ 0 & Z \end{bmatrix} = \begin {bmatrix} R_A & R_B \\ R_C & R_D\end{bmatrix}^T \begin {bmatrix} A & C \\ 0 & B \end{bmatrix} \begin {bmatrix} R_A & R_B \\ R_C & R_D \end{bmatrix}$

where $X$ and $Z$ are upper triangular.

$ = \begin {bmatrix} R_A^T & R_C^T \\ R_B^T & R_D^T\end{bmatrix} \begin {bmatrix} A & C \\ 0 & B \end{bmatrix} \begin {bmatrix} R_A & R_B \\ R_C & R_D \end{bmatrix}$

$ = \begin {bmatrix} R_A^T A & R_A^TC + R_C^T B \\ R_B^TA & R_B^TC + R_D^T B\end{bmatrix} \begin {bmatrix} R_A & R_B \\ R_C & R_D \end{bmatrix}$

$ = \begin {bmatrix} R_A^T A R_A + R_A^TCR_C + R_C^T B R_C & R_A^T A R_B + R_A^TCR_D + R_C^T B R_D \\ R_B^T A R_A + R_B^TCR_C + R_D^T B R_C & R_B^T A R_B + R_B^TCR_D + R_D^T B R_D\end{bmatrix}$

So we want to chose R so that:

$\begin{align} \text{Upper Triangular} &= R_A^T A R_A + R_A^TCR_C + R_C^T B R_C \\ \text{Anything} &= R_A^T A R_B + R_A^TCR_D + R_C^T B R_D \\ \text{Zero} &= R_B^T A R_A + R_B^TCR_C + R_D^T B R_C \\ \text{Upper Triangular} &= R_B^T A R_B + R_B^TCR_D + R_D^T B R_D \\ \\ \text{Given that:} \\ \text{Upper Triangular} &= P^TAP \\ \text{Upper Triangular} &= Q^TBQ \\ \end{align}$

You can see by looking at the equations now that you want some terms to be upper triangular and some to be zero. Fit in $R_A = P$, $R_B = 0$, $R_C = 0$, and $R_D = Q$, you get:

$\begin{align} \text{Upper Triangular} &= P^T A P + P^TC0 + 0^T B 0 \\ &= P^T A P \\ \text{Anything} &= P^T A 0 + P^TCQ + 0^T B Q \\ &= P^TCQ \\ \text{Zero} &= 0^T A P + 0^TC0 + Q^T B 0 \\ &= 0 \\ \text{Upper Triangular} &= 0^T A 0 + 0^TCQ + Q^T B Q \\ &= Q^T B Q \\ \end{align}$

$R = \begin {bmatrix} P & 0 \\ 0 & Q \end{bmatrix}$