 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2021/prelims/mathematics-i/4/b/iv/solution

If every

∥ v_{i} ∥ = 1

, then each diagonal entry of the Gram matrix is

A_{ii} = ⟨ v_{i}, v_{i} ⟩ = 1

, so its matrix trace is

n

0 \leq det A = \prod_{i = 1}^{n} λ_{i} \leq (\frac{\sum _{i = 1}^{n} λ _{i}}{n})^{n} = (\frac{tr A}{n})^{n} = 1 .

The bound is attained by any orthonormal family, whose Gram matrix is the identity matrix.

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