# Matrix equations motivated by generalization of $QR$ decomposition

The following problem is motivated by considering a certain generalization of the $$QR$$ decomposition of a matrix.

Let $$A, B \in M_n(\mathbb{R})$$.

(i) Can we always find $$Q_1, Q_2 \in M_n(\mathbb{R})$$ orthogonal and $$R_1, R_2 \in M_n(\mathbb{R})$$ upper-triangular such that

$$\begin{equation} (*) \quad A = Q_1 R_1 + Q_2 R_2 \quad \mbox{and}\quad B = Q_1 R_2 + Q_2 R_1? \end{equation}$$

(ii) Can we always find $$Q_1, Q_2 \in M_n(\mathbb{R})$$ such that $$\begin{equation} Q_1^T Q_1 + Q_2^TQ_2 = I_n, \quad Q_1^TQ_2 = -Q_2^T Q_1 \end{equation}$$ and $$R_1, R_2 \in M_n(\mathbb{R})$$ upper-triangular for which the equations given by (*) hold?

• Do you mean $B = Q_1 R_2 + Q_2 R_1$? – Igor Khavkine Dec 5 '19 at 8:20
• Yes, that was a typo. Corrected it now. Thanks. – Math Tourist 9000 Dec 5 '19 at 15:31
• I've added an additional sub-question to the problem statement. – Math Tourist 9000 Dec 5 '19 at 16:14
• For orthogonal matrices $Q^TQ=I$ so your first equation in (ii) can never be satisfied. – Alexandre Eremenko Dec 5 '19 at 20:37
• Yes, that is why in (ii) there was no condition for $Q_1$ and $Q_2$ to be orthogonal. – Math Tourist 9000 Dec 5 '19 at 21:08