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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 40A / a / Solution

The power method starts from

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.

Inverse iteration with shift

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

.

Solved by gpt-5.6-sol high.

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