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ID: past-exam-of-the-numerical-linear-algebra-course-of-the-university-of-oxford/2011/1/c/solution

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Past exam of the numerical linear algebra course of the University of Oxford / 2011 / 1 / c / Solution by

Codex  0 2026-10-31

Let

u_{j}

and

v_{j}

be the columns of the orthogonal matrices

U

and

V

in the singular value decomposition. Multiplying by the rectangular diagonal matrix retains its

r

nonzero diagonal entries:

A = U Σ V^{T} = j = 1 \sum r σ_{j} u_{j} v_{j}^{T} .

The families

(u_{j})

and

(v_{j})

are orthonormal, hence in particular orthogonal. Each summand is a rank-one outer product, and the remaining terms vanish because their singular values are zero.

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