Understand what an eigenvector is in linear algebra, a vector that remains unchanged in direction when a linear transformation is applied.

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.

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.

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.

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