Obtaining the QR Decomposition by Pairs of Row and Column Operations

Abstract

Let S = {vl, 2, ..., Vm } be a set of linearly independent vectors in Rn and let A be the matrix that has these vectors as its columns. The Gram-Schmidt process can be applied to the column vectors to produce a new matrix whose column vectors are orthonormal and whose column space is the same as A's. The process replaces each column of A by a linear combination of that column and its predecessors. If Q denotes the matrix with orthonormal columns, then A = QR, where R is an upper-triangular nonsingular matrix. This is the factorization or of A. In this note, we show how to obtain the QR decomposition by using pairs of row and column operations. Suppose the n by m matrix A = [aij] has linearly independent columns vj (alj, a2j, ..., anj)T for j = 1, 2,..., m. Then A = [aij] for i = 1, 2,..., n. Since ATA is symmetric, and its diagonal elements are positive, we can use n(n 1)/2 pairs of row and column operations on ATA to annihilate all of its off-diagonal entries. That is, we obtain BTATAB = diag[dl, d2,..., dm], where B and BT are products of respective lower-triangular and upper-triangular matrices. Since BTATAB = (AB)T(AB) = diag[dl, d2, ..., dm], the columns of AB are orthonormal. Letting C be a square root of this diagonal matrix, we have (C-1BTAT)(ABC -) = I. Thus, the matrix Q = ABC-~ has orthonormal columns. So A = QR, where R = CB-1 is an upper-triangular nonsingular matrix.