What is an Eigenvector?

An eigenvector is a fundamental concept in linear algebra that represents a vector which remains unchanged in direction when a linear transformation is applied to it. Simply put, when a matrix transformation is performed on an eigenvector, the resulting vector points in the same direction as the original one, although it may be scaled by a factor known as the eigenvalue.

Eigenvector Definition and Eigenvalue Problem

Mathematically, an eigenvector v of a linear transformation T satisfies the equation T(v) = λv, where λ is a scalar value known as the eigenvalue corresponding to that eigenvector. The eigenvalue problem is finding the eigenvectors and eigenvalues of a given matrix, which is crucial in various applications in physics, engineering, and computer science.

Matrix Eigenvectors and Applications

In matrix form, the eigenvectors are determined by solving the characteristic equation det(A - λI) = 0, where A is the matrix, λ is the eigenvalue, and I is the identity matrix. Eigenvectors have significant applications such as in principal component analysis, graph theory, quantum mechanics, and image processing.