Eigenvalues
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.
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:
- In physics, they are used in the study of vibrations, quantum mechanics, and stability theory.
- In computer science, they are used in computer graphics, machine learning, and Google's PageRank algorithm.
- In economics, they are used in input-output models, game theory, and Markov chains.