Note that this question is also posted in mathoverflow. I'm new to the community so I'm also posting this here if it is more appropriate.
I am given a square matrix $A = [A_1, A_2, ..., A_n]$, where columns $A_i$ are orthogonal to each other. In another word, $A = OD$, where $O$ is an orthogonal matrix and $D$ is a diagonal matrix. The target is to decompose $A$ into a product of two matrices of the same type. That is, $A = X_1X_2$, where $X_1$ and $X_2$ are both named 'xxx'.
('The same type' here just means that we can classify them into one specific category. I know this is pretty vague, but I didn't come up with a better explanation. E.g, we can decompose $A = O \lambda$, where $\lambda$ is a scalar, as a product of two Jacobian matrices of conformal maps. But this is still a special case of $A$.)
For example, if we consider a special case of $A$ as an orthogonal matrix (i.e., $D = I$), one can simply decompose $A$ as a product of two orthogonal matrices. However, this is a very limited case. I am wondering whether there is a general decomposition of $A$?
Any suggestion would be greatly appreciated.
Thanks to @BenGrossmann for making the question clearer: I'm looking for some "type of matrix" such that if $P$, $Q$ are matrices of this type, then $PQ$ will necessarily have orthogonal columns.