Past exam of the numerical linear algebra course of the University of Oxford / 2025 / 1 / a / iv

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- Solution 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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