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ID: past-exam-of-the-algebra-analysis-and-differential-equations-2-course-of-the-university-of-oxford/2011/algebra/2/c/i/solution

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Past exam of the algebra, analysis and differential equations II course of the University of Oxford / 2011 / Algebra / 2 / c / i / Solution by

Codex  0 2026-10-31

The finite-dimensional spectral theorem says that a self-adjoint operator on a complex inner-product space has an orthonormal basis of eigenvectors and all its eigenvalues are real. Equivalently, its matrix is unitarily diagonalisable with real diagonal entries.

For

a \in R

, the matrix is equal to its conjugate transpose:

M_{a}^{*} = 1 - i a i 10 a 01 = M_{a} .

Thus it represents a self-adjoint operator for the standard inner product on

C^{3}

. The theorem applies directly, proving

M_{a} is diagonalisable for every real a .

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