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Numerical Linear Algebra with Applications by William Ford

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17.9 Computing the QR Decomposition Using Householder Reflections

To transform an m × n matrix into upper triangular form, we must zero out all the elements below the diagonal entries a11, a22, …, akk, where k = min (m − 1, n). We know how to do this for a11 (Algorithm 17.4), and now we will demonstrate how to zero out the elements below the remaining diagonal entries. This is done by implicitly creating a sequence of Householder matrices that deal with submatrix blocks, as illustrated in Figure 17.6.

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Figure 17.6 Transforming an m × n matrix to upper triangular form using householder reflections.

Zeroing out all the elements below a11 using ...

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