Solution

The power method starts with a unit vector

x_{0}

and iterates

x_{k + 1} = \frac{A x _{k}}{∥ A x _{k} ∥} .

Its eigenvalue estimate is the Rayleigh quotient

r (x_{k}) = \frac{x _{k}^{T} A x _{k}}{x _{k}^{T} x _{k}} .

By the real spectral theorem, choose an orthonormal eigenbasis

v_{1}, \dots, v_{n}

and write

x_{0} = \sum_{j = 1}^{n} c_{j} v_{j} .

Assume

∣ λ_{1} ∣ > ∣ λ_{2} ∣

λ_{1} \neq = 0

, and

c_{1} \neq = 0

. Before normalization,

A^{k} x_{0} = c_{1} λ_{1}^{k} [v_{1} + \sum_{j = 2}^{n} \frac{c _{j}}{c _{1}} (\frac{λ _{j}}{λ _{1}})^{k} v_{j}] .

Thus the direction error is

O (∣ λ_{2} / λ_{1} ∣^{k})

. Orthogonality makes the linear terms vanish from the Rayleigh quotient:

r (x_{k}) - λ_{1} = \frac{\sum _{j = 2}^{n} c _{j}^{2} λ _{j}^{2 k} ( λ _{j} - λ _{1} )}{\sum _{j = 1}^{n} c _{j}^{2} λ _{j}^{2 k}} .

Consequently

r (x_{k}) - λ_{1} = O (\frac{λ _{2}}{λ _{1}}^{2 k}) .

Solved by gpt-5.6-sol high.

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