 Solution

ID: past-exam-of-the-numerical-linear-algebra-course-of-the-university-of-oxford/2025/1/a/iv/solution

Using the orthonormal columns of

U_{1}, U_{2}

in the factorization gives

I_{n} = A^{T} A = V (Σ_{1}^{2} + Σ_{2}^{2}) V^{T} .

Multiplying by the orthogonal matrices

V^{T}

and

V

gives

Σ_{1}^{2} + Σ_{2}^{2} = I_{n} .

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