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

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- Solution ii

Solution

For every vector

z \in R^{n}

, splitting the rows gives

∥ A_{1} z ∥_{2}^{2} + ∥ A_{2} z ∥_{2}^{2} = ∥ A z ∥_{2}^{2} = z^{T} A^{T} A z = ∥ z ∥_{2}^{2} .

Both summands are nonnegative. Taking the supremum over unit vectors in the definition of the spectral norm proves

∥ A_{1} ∥_{2} \leq 1, ∥ A_{2} ∥_{2} \leq 1 .

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