# Eigenvalues and eigenvectors

## Vectors that map to their scalar multiples, and the associated scalars / From Wikipedia, the free encyclopedia

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In linear algebra, an **eigenvector** (/ˈaɪɡənˌvɛktər/) or **characteristic vector** of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding **eigenvalue**, often denoted by $\lambda$, is the factor by which the eigenvector is scaled.

Geometrically, a transformation matrix rotates, stretches, or shears the vectors it acts upon. An eigenvector, corresponding to a real nonzero eigenvalue for that matrix, points in a direction in which it is stretched by the transformation, and is neither rotated nor sheared. The eigenvalue is the factor by which an eigenvector is stretched. If the eigenvalue is negative, the direction is reversed.[1]