triangular/orthogonal matrix properties

Question

Assume A and B are square orthogonal matrices, C and D are square upper triangular matrices with positive diagonals. A, B, C, D all have same dimensions (Q x Q). If AC=BD, then why does A=B and C=D?

Answer 1

By QR decomposition, matrix $S = QR$ where $Q$ is orthogonal and $R$ is upper triangular. If $S$ is invertible, then the factorization is unique if we require that the diagonal elements of $R$ be positive.

So applying this to your question $S_1 = AC$, $S_2 = BD$ and $S_1 = S_2 = S$:

If $S$ are invertible, then $A = B$ and $C = D$ if the diagonal entries of $C$ and $D$ are positive.