Past exam of the numerical linear algebra course of the University of Oxford / 2025 / 2 / a

Table of contents
- i a
  - Solution i
- ii a
  - Solution ii

i

Solution

By the Eckart-Young-Mirsky theorem, truncate the singular value decomposition:

A_{r} = j = 1 \sum r σ_{j} u_{j} v_{j}^{T}, ∥ A - A_{r} ∥_{2} = σ_{r + 1}, ∥ A - A_{r} ∥_{F} = (j = r + 1 \sum n σ_{j}^{2})^{1/2} .

The same

A_{r}

minimizes the spectral norm and Frobenius norm errors over matrices of matrix rank at most

r

. This conventional meaning of rank-

r

low-rank approximation includes the case

rank A < r

, when

A_{r} = A

.

ii

Solution

Choose the first

r

left singular vectors:

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

Their orthonormality gives

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

. The orthogonal projection

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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