Make power iteration converge on a tied pair

Adam Zettel · September 25, 2026 · written with an agent

A = (0220), λ = \pm 2 : v_{k + 1} = \frac{( A + s I ) v _{k}}{∥ ( A + s I ) v _{k} ∥}

Power iteration multiplies a vector by $A$ again and again and finds the eigenvalue of largest magnitude. The matrix above has two, $+ 2$ and $- 2$ , of the same magnitude, and the iterate swaps its two entries every step forever. Write shifted_power so that the shift $s$ breaks the tie: iterate with $A + s I$ , whose eigenvalues are $λ + s$ .

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Write $v_{0}$ in the eigenvectors, $v_{0} = a u_{+} + b u_{-}$ . After $k$ steps $A^{k} v_{0} = 2^{k} (a u_{+} + (- 1)^{k} b u_{-})$ : the two parts stay the same size, and the sign of one flips each step. From $(1, 0.3)$ the normalized iterate alternates between $(0.958, 0.287)$ and $(0.287, 0.958)$ , and its Rayleigh quotient $v^{T} A v$ is 1.101, which is neither eigenvalue.

$A + s I$ has the same eigenvectors, with eigenvalues $2 + s$ and $- 2 + s$ . With $s = 1$ those are 3 and $- 1$ , so the $u_{+}$ part wins by a factor of 3 per step; with $s = - 1$ they are 1 and $- 3$ , and $u_{-}$ wins. The Rayleigh quotient of the original $A$ then gives the eigenvalue itself.

The second matrix, $(12 2 - 1)$ , has eigenvalues $\pm 5$ , again a tie. The expected vectors are numpy.linalg.eigh's (posts/refs/make-power-iteration-converge-on-a-tied-pair.py), signed so that the first entry is positive; the function must return its vector signed the same way.

Your task

Write these 2 functions:

shifted_power(A: Arr, s, iters): The unit vector power iteration with A + sI reaches after iters steps from (1, 0.3), signed so its first entry is positive. As written it ignores the shift s.
rayleigh(A: Arr, v: Arr): The Rayleigh quotient v^{\mathsf T} A v of a unit vector v.

Make these tests pass 4 of 4 fail

shift +1 finds eigenvalue 2, vector (1, 1) / sqrt 2 assert_near: |0.957826 - 0.707107| = 0.25072 > 1e-12 (and 2 more) (fails)
shift -1 finds eigenvalue -2, vector (1, -1) / sqrt 2 assert_near: |0.957826 - 0.707107| = 0.25072 > 1e-12 (and 2 more) (fails)
on [[1, 2], [2, -1]] shift +1 finds sqrt 5 = 2.2360680 with numpy's vector assert_near: |0.957826 - 0.850651| = 0.107175 > 1e-12 (and 2 more) (fails)
and shift -1 finds -sqrt 5 assert_near: |0.957826 - 0.525731| = 0.432095 > 1e-12 (and 2 more) (fails)

Printed

v_plain: [0.957826, 0.287348]

1.10092

shifted_eigs: [3, -1, 1, -3]

Your task

Solutions 0