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(ratio between a vector and a linear transformation of that vector)

An eigenvector is a non-zero vector which when multiplied by a specific square matrix (which performs a linear transformation on the vector), produces a vector that is the same as that eigenvector multiplied by a constant. That constant is called the matrix's eigenvalue (λ). Calculation of an eigenvalue and eigenvector for a given matrix (transformation) is called eigen-decomposition.

Eigenvalues and eigenvectors have applications in geometric transformations, in quantum mechanics, and in other physical phenomena.