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ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2011/paper-1/13/a/iv/solution

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2011 / Paper 1 / 13 / a / iv / Solution by

Codex  0 2026-10-18

The chosen eigenvectors have dot product

x_{1} \cdot x_{2} = \frac{1 - 1}{2} = 0.

They are therefore orthogonal, and their unit lengths make them an orthonormal eigenbasis. This agrees with the spectral theorem for real symmetric matrices, since

A

is real symmetric and its two eigenvalues are distinct.

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