Eigenvalues and eigenvectors
Introduction
Eigenvalues and eigenvectors are fundamental concepts in linear algebra, with applications in various fields such as physics, computer science, engineering, and economics. They are used to simplify complex problems, allowing for easier computation and analysis.
Definition
An eigenvalue is a scalar associated with a given 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.