Eigenvalues

From Canonica AI

Introduction

In the field of linear algebra, an eigenvalue is a scalar associated with a given linear transformation of a vector space. The concept of eigenvalues is central to many areas of mathematics and its applications, including differential equations, quantum mechanics, and machine learning.

Definition

Given a linear transformation represented by a square matrix A, a scalar λ is an eigenvalue of A if there exists a non-zero vector v, such that Av = λv. This vector v is called an eigenvector of A corresponding to the eigenvalue λ.

A square matrix with a vector and a scalar, illustrating the concept of eigenvalues and eigenvectors.
A square matrix with a vector and a scalar, illustrating the concept of eigenvalues and eigenvectors.

Properties

Eigenvalues have several important properties:

  • The sum of the eigenvalues of a matrix, counted with their multiplicities, equals the trace of the matrix.
  • The product of the eigenvalues of a matrix, counted with their multiplicities, equals the determinant of the matrix.
  • The eigenvalues of a diagonal or triangular matrix are its diagonal entries.
  • If a matrix is orthogonal, then its eigenvalues are complex numbers of absolute value 1.

Calculation

Eigenvalues of a matrix can be calculated by solving the characteristic equation, which is derived from the equation Av = λv. The characteristic equation is given by det(A - λI) = 0, where I is the identity matrix of the same size as A, and det denotes the determinant.

Applications

Eigenvalues and eigenvectors have wide-ranging applications in various fields:

See Also

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