 Solution

ID: past-exam-of-the-numerical-linear-algebra-course-of-the-university-of-oxford/2011/1/e/solution

The two columns are orthogonal, with lengths one and two; equivalently

A^{T} A = diag (1, 4)

. Ordering the singular values decreasingly requires swapping the right coordinate directions. One full singular value decomposition is

U = 100 00 - 1 010, Σ = 200010, V = (0110) .

Both

U

and

V

U Σ V^{T} = A

. Hence

∥ A ∥_{2} = 2, ∥ A ∥_{F} = 5 .

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