Past exam of the numerical linear algebra and continuous optimisation course of the University of Oxford / 2010 / A / 3 / a

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Solution

The off-diagonal Toeplitz matrix

T = A - I

is symmetric and has row and column absolute sums bounded by

r = 2 \sum_{j = 2}^{m} ∣ t_{j} ∣ \leq \frac{2}{3} .

For an elementary Euclidean bound, apply the weighted Cauchy-Schwarz inequality in each row:

∣ (T x)_{i} ∣^{2} \leq (\sum_{j} ∣ T_{ij} ∣) (\sum_{j} ∣ T_{ij} ∣ ∣ x_{j} ∣^{2}) .

Sum over

i

and use both absolute-sum bounds to get

∥ T x ∥_{2}^{2} \leq r^{2} ∥ x ∥_{2}^{2}

. Therefore

∥ T ∥_{2} \leq 2 j = 2 \sum m ∣ t_{j} ∣ \leq \frac{2}{3} .

The entry

t_{1}

is unused because the diagonal of

A

is fixed to one; including it only weakens the bound. This is the row-column bound for the matrix 2-norm.

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