View source for Eigenvalues and eigenvectors

== Introduction ==
Eigenvalues and eigenvectors are fundamental concepts in [[Linear Algebra|linear algebra]], with applications in various fields such as [[Physics|physics]], [[Computer Science|computer science]], [[Engineering|engineering]], and [[Economics|economics]]. They are used to simplify complex problems, allowing for easier computation and analysis.

[[Image:Detail-35735.jpg|thumb|center|A matrix with its corresponding eigenvalues and eigenvectors|class=only_on_mobile]]
[[Image:Detail-35736.jpg|thumb|center|A matrix with its corresponding eigenvalues and eigenvectors|class=only_on_desktop]]

== Definition ==
An eigenvalue is a scalar associated with a given [[Linear Transformation|linear transformation]] that characterizes the magnitude by which a corresponding eigenvector is scaled during the transformation. An eigenvector, on the other hand, is a non-zero vector that only changes by a scalar factor when a linear transformation is applied to it.

== Mathematical Representation ==
Given a square matrix A and a vector v, if the multiplication of A and v yields a scalar multiple of v, then v is an eigenvector of A, and the scalar is the corresponding eigenvalue. This relationship is represented mathematically as:

Av = λv

where:
* A is a square matrix
* v is an eigenvector of A
* λ (lambda) is the corresponding eigenvalue

== Calculation ==
The eigenvalues of a matrix A are calculated by solving the characteristic equation, which is obtained by setting the determinant of the matrix subtracted by λ times the identity matrix equal to zero. The eigenvectors are then calculated by substituting each eigenvalue back into the equation (A - λI)v = 0 and solving for v.

== Properties ==
Eigenvalues and eigenvectors have several important properties. For instance, a matrix and its transpose have the same eigenvalues. The sum of the eigenvalues equals the trace of the matrix, and the product of the eigenvalues equals the determinant of the matrix. 

== Applications ==
Eigenvalues and eigenvectors have wide-ranging applications. In physics, they are used in quantum mechanics to represent observable quantities. In computer science, they are used in algorithms for image recognition and machine learning. In engineering, they are used in the analysis of systems of differential equations. In economics, they are used in the analysis of multi-sector economic models.

== See Also ==
* [[Matrix (mathematics)|Matrix]]
* [[Linear Algebra]]
* [[Differential Equations]]
* [[Quantum Mechanics]]

[[Category:Mathematics]]
[[Category:Linear Algebra]]
[[Category:Applied Mathematics]]