# What's the formula? - eigenvectors from eigenvalues

• Stephen Tashi
In summary, the conversation discusses a mathematical discovery about using the eigenvalues and eigenvectors of submatrices to solve for the eigenvectors of a larger matrix. This method reduces the complexity of solving for eigenvectors and an example can be found in the graphic provided by the article.
The idea is that the eigenvalues of a matrix A combined with the eigenvalues and eigenvectors of the submatrices of A can be used to compute the eigenvectors of A.

It reduces the complexity of eigenvector solving for large matrices but reducing the problem to smaller matrices and using the computed eigenvalues to aid in the solution.

I still haven't seen a numerical example though the graphic in The Atlantic article gives you an example to be completed by the reader.

## 1. What is the formula for finding eigenvectors from eigenvalues?

The formula for finding eigenvectors from eigenvalues is: Ax = λx, where A represents the matrix, λ represents the eigenvalue, and x represents the eigenvector.

## 2. Why is finding eigenvectors from eigenvalues important?

Finding eigenvectors from eigenvalues is important because it allows us to understand the behavior of a linear transformation. Eigenvectors are used in various applications such as image processing, data compression, and machine learning.

## 3. What information can we gather from eigenvectors and eigenvalues?

Eigenvectors and eigenvalues provide information about the magnitude and direction of a transformation. They also help determine the stability and behavior of a system.

## 4. How do we solve for eigenvectors and eigenvalues?

To solve for eigenvectors and eigenvalues, we can use various methods such as the power method, inverse iteration, or the QR algorithm. These methods involve finding the characteristic polynomial of the matrix and solving for its roots.

## 5. Can the formula for finding eigenvectors from eigenvalues be applied to any matrix?

Yes, the formula for finding eigenvectors from eigenvalues can be applied to any square matrix. However, for non-square matrices, we use singular value decomposition to find the eigenvectors and eigenvalues.

Replies
4
Views
1K
Replies
9
Views
2K
Replies
11
Views
1K
Replies
1
Views
1K
Replies
19
Views
3K
Replies
4
Views
2K
Replies
3
Views
1K
Replies
1
Views
1K
Replies
2
Views
1K
Replies
15
Views
2K