# 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:

* <mark style="color:red;">v is a non-zero vector called the eigenvector associated with the eigenvalue λ.</mark> In other words, when <mark style="color:green;">**a matrix A operates on its eigenvector, the result is a scaled version of the eigenvector itself.**</mark>

### The important properties of eigenvalues and eigenvectors

1. <mark style="color:green;">**Multiplicity**</mark>: An eigenvalue may have a multiplicity greater than 1, which indicates that there are multiple linearly independent eigenvectors associated with that eigenvalue.
2. <mark style="color:green;">**Eigenvalue equation**</mark>: 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. <mark style="color:green;">**Determinant and trace**</mark>: 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. <mark style="color:green;">**Diagonalizability**</mark>: 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.


---

# Agent Instructions
This documentation is published with GitBook. GitBook is the documentation platform designed so that both humans and AI agents can read, navigate, and reason over technical content effectively. Learn more at gitbook.com.

## Querying This Documentation
If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question.

Perform an HTTP GET request on the current page URL with the `ask` query parameter:

```
GET https://aisuko.gitbook.io/wiki/ai-techniques/adaptation/lora/svd/eigenvalues-of-a-covariance-matrix/eigenvalues.md?ask=<question>
```

The question should be specific, self-contained, and written in natural language.
The response will contain a direct answer to the question and relevant excerpts and sources from the documentation.

Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.