# Matrix A and its inverse have the same eigenvectors

• Mr Davis 97
In summary, the conversation discusses whether each eigenvector of an invertible matrix A is also an eigenvector of its inverse, A-1. It is shown that if A is invertible and has distinct eigenvalues, then A and its inverse have the same eigenvectors, with the eigenvalues being inverse as well.
Mr Davis 97

## Homework Statement

T/F: Each eigenvector of an invertible matrix A is also an eignevector of A-1

## 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}##?

Mr Davis 97 said:

## Homework Statement

T/F: Each eigenvector of an invertible matrix A is also an eignevector of A-1

## 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}##?
It's ##A^{-1}=(PDP^{-1})^{-1}=PD^{-1}P^{-1}##. You have provided the argument that is needed, and you showed, that the eigenvalues are inverse, too, e.g. the elements in ##D## and ##D^{-1}##. Why should there be a discrepancy?

## 1. What is an eigenvector?

An eigenvector is a vector that, when multiplied by a square matrix, results in a multiple of the original vector. The multiple is known as the eigenvalue.

## 2. How is the inverse of a matrix related to its eigenvectors?

Matrix A and its inverse have the same eigenvectors, meaning that the eigenvectors of A are also the eigenvectors of A-1. This is because the eigenvectors are the same regardless of whether the matrix is multiplied by its inverse or not.

## 3. What is the significance of having the same eigenvectors for a matrix and its inverse?

Having the same eigenvectors for a matrix and its inverse is significant because it means that the eigenvectors are preserved in the transformation, making it easier to calculate the eigenvalues and diagonalize the matrix.

## 4. Can a matrix have different eigenvalues for itself and its inverse?

Yes, a matrix can have different eigenvalues for itself and its inverse. The eigenvalues of a matrix and its inverse are related by the reciprocal of the original eigenvalues. Therefore, even if the eigenvectors are the same, the eigenvalues may be different.

## 5. How can the eigenvectors of a matrix and its inverse be used in real-world applications?

The eigenvectors of a matrix and its inverse can be used in various applications, such as image processing, data compression, and solving systems of differential equations. By diagonalizing the matrix using its eigenvectors, complex calculations can be simplified and made more efficient.

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