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?

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

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