Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 31L / b

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- Solution b

Solution

For a positive semidefinite matrix, the sum of the eigenvalues is its trace. Thus

S = {M = M^{T} : M ⪰ 0, tr M \leq s} .

If

M, N \in S

and

0 \leq t \leq 1

, then

tM + (1 - t) N ⪰ 0

and

tr (tM + (1 - t) N) = t tr M + (1 - t) tr N \leq s .

Hence $S$ is convex.

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