 Solution

ID: past-exam-of-the-numerical-linear-algebra-course-of-the-university-of-oxford/2025/1/b/i/solution

Choose the nonnegative square roots of the eigenvalues of

A^{T} A

for the diagonal entries of

Σ

, in nonincreasing order. Its assumed matrix inverse exists, so these entries are positive. The eigendecomposition gives

U^{T} U = Σ^{- 1} V^{T} A^{T} A V Σ^{- 1} = Σ^{- 1} Σ^{2} Σ^{- 1} = I_{n} .

Hence

U

has orthonormal columns. Also

U Σ V^{T} = A V Σ^{- 1} Σ V^{T} = A .

Together with the orthogonal matrix

V

and the nonnegative diagonal matrix

Σ

, these are precisely the properties of a thin singular value decomposition.

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