 Solution

ID: past-exam-of-the-numerical-linear-algebra-and-continuous-optimisation-course-of-the-university-of-oxford/2010/a/3/b/solution

x^{T} A x > 0

for every nonzero real vector

x

. The preceding matrix 2-norm bound gives

x^{T} A x = ∥ x ∥_{2}^{2} + x^{T} T x \geq ∥ x ∥_{2}^{2} - ∥ x ∥_{2} ∥ T x ∥_{2} \geq \frac{1}{3} ∥ x ∥_{2}^{2} > 0 .

Thus

A

is a symmetric positive-definite matrix, independently of the detailed positions of the nonzero entries of

t

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