 Solution

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

A = U Σ V^{T}, U \in R^{m \times m}, V \in R^{n \times n}, U^{T} U = I_{m}, V^{T} V = I_{n},

where

Σ \in R^{m \times n}

has only diagonal entries, with

σ_{1} \geq \dots \geq σ_{m i n (m, n)} \geq 0

. These are the singular values. The number of positive singular values is the matrix rank. The columns of

V

U

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