What is an inverse matrix's eigenvalue?

A matrix A has an eigenvalue λ only if A−1 has eigenvalue λ−1. For example: Av=λv⇒A−1Av=λA−1v⇒A−1v=1λv If matrix A has eigenvalue λ, then I-A has eigenvalue 1−λ. Therefore, (I−A)−1 has eigenvalue 11−λ

Step by step solution to "What is an inverse matrix's eigenvalue?"

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Algebra
Algebra I
Linear equations and graphs

tugmiddelc0

Answered question

2021-11-20

What is an inverse matrix's eigenvalue?

Answer & Explanation

barcelodurazo0q

Beginner2021-11-21Added 13 answers

A matrix A has an eigenvalue $λ$ only if $A^{- 1}$ has eigenvalue $λ^{- 1}$ . For example:
$A v = λ v \Rightarrow A^{- 1} A v = λ A^{- 1} v \Rightarrow A^{- 1} v = \frac{1}{λ} v$
If matrix A has eigenvalue $λ$ , then I-A has eigenvalue $1 - λ$ . Therefore, ${(I - A)}^{- 1}$ has eigenvalue $\frac{1}{1 - λ}$

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 $1 - λ$ , its inverse (I−A)−1, in fact for any rational function of A (if well defined; this is where you need $λ 1 < 1$ ).

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