Given two or more semi-orthogonal matrices, is there any way one could combine them to obtain a new semi-orthogonal matrix? To be specific, the original matrices have orthogonal rows only.
Obviously adding or subtracting them wouldn't do the trick. Nor does an addition/subtraction of their singular values. But that's about as far as my matrix algebra goes, so I'm at my absolute wit's end.
I've checked out a couple of textbooks too but it's so difficult to find information on these matrices.
I'd be grateful for any suggestions, let alone solutions, so please chime in even if you're unsure of your solution. Thanks in advance.
Addendum: For context, I am dealing with sparse semi orthogonal matrices where every column has only one non-zero element. My aim is to somehow merge two such matrices to create a new matrix of the same type(where every column has exactly one non-zero element).
Start with a matrix $S$ that's $n \times k$, where $n \le k$.
You could unit-ize the rows, i.e., replace the $i$th row, $r_i$, with $r_i / \| r_i\|$. That'd still be semi-orthogonal.
Having done item 1, you could add rows to the matrix to get a new matrix $U$ whose rows are all orthogonal and unit length (using the Gram-Schmidt process). $U$ will still be semi-orthgonal, but also orthogonal.
If you do items 1 and 2 to two $n \times k$ semi-orthog matrices $A$ and $B$ to produce $k \times k$ matrices $U$ and $V$, then $UV$ and $VU$ will be orthogonal as well (hence semi-orthogonal), and removing $k-n$ rows from each will give you semi-orthogonal matrices of the same size as $A$ and $B$.
Clearly I can diddle around with this endlessly, converting to orthogonal, and then back again. But what's the point? Without knowing something about the goal, it's hard to know.