 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ii/paper-4/40a/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

q_{0} \neq = 0

and iterates

q_{j + 1} = \frac{A q _{j}}{∥ A q _{j} ∥} .

If the eigenvalue of largest modulus is separated by a spectral gap and the initial vector is nonorthogonal to a corresponding eigenvector, the iterates converge in direction to that eigenvector.

s

instead solves

(A - s I) y_{j + 1} = q_{j}, q_{j + 1} = \frac{y _{j + 1}}{∥ y _{j + 1} ∥} .

It applies the power method to

(A - s I)^{- 1}

and therefore finds an eigenvector whose eigenvalue is closest to

s

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