- #1

Mr Davis 97

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## Homework Statement

T/F: Each eigenvector of an invertible matrix A is also an eignevector of A

^{-1}

## Homework Equations

## The Attempt at a Solution

I know that if A is invertible and ##A\vec{v} = \lambda \vec{v}##, then ##A^{-1} \vec{v} = \frac{1}{\lambda} \vec{v}##, which seems to imply that A and its inverse have the same eigenvectors. However, what about if A has all distinct eigenvalues, then ##A = PDP^{-1}##. From this, we conclude that ##A^{-1} = P^{-1} D^{-1} P##. Doesn't this show that A and its inverse have different eigenvectors? Since for A the matrix of eigenvectors is ##P## while for A

^{-1}the matrix of eigenvectors is ##P^{-1}##?