Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 1 / 8F / b / i

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- Solution i

Solution

If

A v = λ v

for a nonzero real vector

v

, then skew-symmetry gives

λ ∥ v ∥^{2} = v^{T} A v = - (A v)^{T} v = - λ ∥ v ∥^{2} .

Thus

λ = 0,

so a real skew-symmetric matrix has no nonzero real eigenvalue.

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