 Solution

ID: past-exam-of-the-numerical-linear-algebra-course-of-the-university-of-oxford/2025/2/a/ii/solution

Choose the first

r

Q_{*} = [u_{1}, \dots, u_{r}] .

Q_{*}^{T} Q_{*} = I_{r}

Q_{*} Q_{*}^{T}

fixes

u_{j}

for

j \leq r

and annihilates

u_{j}

for

j > r

. Applying it to the singular value decomposition therefore gives

Q_{*} Q_{*}^{T} A = \sum_{j = 1}^{r} σ_{j} u_{j} v_{j}^{T} = A_{r} .

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