Eigenvalues and eigenvectors

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 ${\displaystyle \lambda }$, 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]