🧧Eigenvalues

What is it?

Eigenvalues are a fundamental concept in linear algebra.

Formula

Given a square matrix A, an eigenvalue is a scalar λ that satisfies the equation:

$$A*v=λ * v$$

Here:

• v is a non-zero vector called the eigenvector associated with the eigenvalue λ. In other words, when a matrix A operates on its eigenvector, the result is a scaled version of the eigenvector itself.

The important properties of eigenvalues and eigenvectors

1. Multiplicity: An eigenvalue may have a multiplicity greater than 1, which indicates that there are multiple linearly independent eigenvectors associated with that eigenvalue.

2. Eigenvalue equation: The equation A * v = λ * v can be rewritten as (A - λI) * v = 0, where I is the identity matrix. This equation implies that the matrix (A - λI) is singular, i.e., it has a determinant of zero.

3. Determinant and trace: The sum of the eigenvalues of a matrix A is equal to its trace (the sum of the diagonal elements) and the product of its eigenvalues is equal to its determinant.

4. Diagonalizability: A matrix A is said to be diagonalizable if it has a full set of linearly independent eigenvectors. In this case, A can be expressed as A = P * D * P^(-1), where P is a matrix consisting of the eigenvectors of A, and D is a diagonal matrix with the corresponding eigenvalues on its diagonal.

Eigenvalues have numerous applications in various fields

• In quantum mechanics, eigenvalues and eigenvectors play a fundamental role in determining the possible energy levels and corresponding states of a quantum system.

• Eigenvalue decomposition is used in image processing for tasks like image compression and denoising.