In [Chahlaoui, Gallivan and Van Dooren, 2004] a recursive procedure is designed for computing an approximation of the left and right dominant singular subspaces of a matrix, whose columns are produced incrementally. The method is
particularly suited for matrices with many more rows than columns. The procedure consists of a few steps. In one of these steps a Householder transformation
is multiplied to an upper triangular matrix. The following step consists in recomputing an upper triangular matrix from the latter product. In [Chahlaoui, Gallivan and Van Dooren, 2004] it is said that the latter step is accomplished
in O(k^3) operations, where k is the order of the triangular matrix. In this short note we show that this step can be accomplished in O(k^2) operations. The note is organized as follows. In Section 2 the algorithm proposed in [Chahlaoui, Gallivan and Van Dooren, 2004] is described and the proposed modiﬁcation is described in Section 3.