The Wikipedia article on QR decomposition states:
More generally, we can factor a complex m×n matrix A, with m ≥ n, as the product of an m×m unitary matrix Q and an m×n upper triangular matrix R.
I was just wondering what would happen if I tried to QR decompose a matrix with more columns than rows. I'd just follow Gram-Schmidt (described in the same Wikipedia article) until I got an $m \times m$ orthonormal matrix. Then, the remaining columns of $R$, (given the original one's were $c_j$) would just become $Q^{-1}c_j$.
I'm convinced now that the only reason they require $m>n$ is for semantic purposes, not wanting to call the resulting $R$ matrix "upper triangular". Because one could always take just the square part of a "fat" matrix, QR decompose it and then multiply the remaining columns by $Q^{-1}$ from the right. We would get something like the picture below.