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In machine learning and linear algebra, an eigenvector is a vector that remains in the same direction after a linear transformation is applied to it. More formally, for a square matrix A and a non-zero vector v, if Av is a scalar multiple of v, then v is an eigenvector of A, and the scalar multiple is called the eigenvalue corresponding to that eigenvector.

Mathematically, if we have a matrix A and an eigenvector v with eigenvalue λ, it can be represented as:

          A * v = λ * v ----------------------------------------------- [3922]

Eigenvectors and eigenvalues are fundamental concepts in linear algebra and have several applications in machine learning, particularly in techniques like Principal Component Analysis (PCA), Singular Value Decomposition (SVD), and various dimensionality reduction and feature extraction methods. They help in reducing the dimensionality of data while preserving important information or in analyzing the dominant modes of variation in a dataset.

In machine learning, eigenvectors and eigenvalues can be used for tasks such as image compression, facial recognition, and understanding the underlying structure of high-dimensional data. They are also essential in solving systems of linear differential equations, which are often used in modeling dynamic systems in various fields of science and engineering.

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