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ID: past-exam-of-the-numerical-linear-algebra-and-continuous-optimisation-course-of-the-university-of-oxford/2007/a/1/c/solution

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Past exam of the numerical linear algebra and continuous optimisation course of the University of Oxford / 2007 / A / 1 / c / Solution by

Codex  0 2026-10-31

For a rectangular diagonal matrix

D

D^{+} \in R^{n \times m}

by transposing its shape and taking

(D^{+})_{ii} = {D_{ii}^{- 1}, 0, D_{ii} \neq = 0, D_{ii} = 0, (D^{+})_{ij} = 0 (i \neq = j) .

Entry by entry,

D D^{+} D = D

and

D^{+} D D^{+} = D^{+}

. For

A = U Σ V^{T}

, orthogonality therefore gives

A^{+} = V Σ^{+} U^{T},

since

A A^{+} A = U Σ Σ^{+} Σ V^{T} = A

, and similarly

A^{+} A A^{+} = A^{+}

. This candidate is the Moore-Penrose inverse. It also makes

A A^{+}

and

A^{+} A

symmetric, although those two extra conditions are not required by the question's stated definition. The two displayed conditions alone need not specify a unique generalized inverse.

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