 Solution

ID: past-exam-of-the-options-ii-course-of-the-university-of-oxford/2013/p4/c/solution

Comparing the

j

th columns of

A = QR

gives

a_{j} = Q r_{j}

. Thus

∥ a_{j} ∥^{2} = r_{j}^{T} Q^{T} Q r_{j} = r_{j}^{T} r_{j} = ∥ r_{j} ∥^{2} .

Taking nonnegative square roots proves

∥ a_{j} ∥ = ∥ r_{j} ∥

. This is simply norm preservation by an orthogonal matrix.

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