How to devise a simple modification of the power method to handle the following
case:
λ1 = -λ2 > |λ3| ≥ |λ4| ≥ |λ5| ≥ ... ≥ |λn| ?
Could you please help me?
2026-03-26 14:31:54.1774535514
Modification of Power Method
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Proceed as usual in the theoretical exploration of the power method: Express the original vector and the dynamic of the power iteration in an eigenbasis of $A$.
$$ x=c_1v_1+c_2v_2+c_3v_3…\\ A^nx=c_1λ_1^nv_1+c_2(-λ_1)^nv_2+c_3λ_3^nv_3+… $$
Comparing $A^nx$ and $A^{n+1}x$ should give you an idea or two.