QR is useful for analysis and solution of underdetermined systems too.
Same approach referenced in Golub & Loan (ch. 5.7.2).
Will test m == n again and will let you know the details tomorrow (now is pretty late in Tokyo).
Btw, Intel compiler bounds checker detects many out-bounds reads & writes in SparseQR even for m > n.
Whereas no such issues are detected for sparse LU & LDLt.
It is again ICC 13.1 & MSVC combo.