What are eigenvalues and eigenvectors?

What are eigenvalues and eigenvectors?

Eigenvalues and eigenvectors are fundamental concepts in linear algebra. Eigenvalues are scalars that indicate how much an eigenvector is stretched or compressed during a linear transformation.

Can you give me an example of how to find them?

Sure! For a matrix A, you find eigenvalues by solving the characteristic equation det(A - λI) = 0, where λ is the eigenvalue and I is the identity matrix.

What does the eigenvector represent in this context?

The eigenvector represents a direction in which the transformation acts by simply stretching or compressing, without changing the direction.

How are they used in real-world applications?

Eigenvalues and eigenvectors are used in various fields, including physics for stability analysis, computer graphics for transformations, and machine learning for dimensionality reduction.