What is an inverse matrix's eigenvalue?

tugmiddelc0

tugmiddelc0

Answered question

2021-11-20

What is an inverse matrix's eigenvalue?

Answer & Explanation

barcelodurazo0q

barcelodurazo0q

Beginner2021-11-21Added 13 answers

A matrix A has an eigenvalue λ only if A1 has eigenvalue λ1. For example:
Av=λvA1Av=λA1vA1v=1λv
If matrix A has eigenvalue λ, then I-A has eigenvalue 1λ. Therefore, (IA)1 has eigenvalue 11λ

Wasither1957

Wasither1957

Beginner2021-11-22Added 17 answers

If you are looking at a single eigenvector v only, with eigenvalue λ, then A just acts as the scalar λ, and any reasonable expression in A acts on v as in λ. This works for expressions I−A, which is really 1−A, so it acts as 1λ, its inverse (I−A)−1, in fact for any rational function of A (if well defined; this is where you need λ1<1).

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