An eigenvector or characteristic vector of a square matrix is a non-zero vector that, when multiplied with , yields a scalar multiple of itself; the scalar multiplier is often denoted by . That is:(Because this equation uses post-multiplication of the matrix A by the vector it describes a right eigenvector.)The number is called the eigenvalue or characteristic value of corresponding to .If two-dimensional space is visualized as a piece of cloth being stretched by the matrix, the eigenvectors would make up the line along the direction the cloth is stretched in and the line of cloth at the center of the stretching, whose direction isn't changed by the stretching either. The eigenvalues for the first line would give the scale to which the cloth is stretched, and for the second line the scale to which it is tightened. A reflection may be viewed as stretching a line to scale −1 while shrinking the axis of reflection to scale 1. For 3D rotations, the eigenvectors form the axis of rotation, and since the scale of the axis is unchanged by the rotation, their eigenvalues are all 1.In analytic geometry, for example, a three-coordinate vector may be seen as an arrow in three-dimensional space starting at the origin. In that case, an eigenvector is an arrow whose direction is either preserved or exactly reversed after multiplication by . The corresponding eigenvalue determines how the length of the arrow is changed by the operation, and whether its direction is reversed or not, determined by whether the eigenvalue is negative or positive.In abstract linear algebra, these concepts are naturally extended to more general situations, where the set of real scalar factors is replaced by any field of scalars (such as algebraic or complex numbers); the set of Cartesian vectors is replaced by any vector space (such as the continuous functions, the polynomials or the trigonometric series), and matrix multiplication is replaced by any linear operator that maps vectors to vectors (such as the derivative from calculus). In such cases, the "vector" in "eigenvector" may be replaced by a more specific term, such as "eigenfunction", "eigenmode", "eigenface", or "eigenstate". Thus, for example, the exponential function is an eigenfunction of the derivative operator, , with eigenvalue , since its derivative is .The set of all eigenvectors of a matrix (or linear operator), each paired with its corresponding eigenvalue, is called the eigensystem of that matrix. Any multiple of an eigenvector is also an eigenvector, with the same eigenvalue. An eigenspace or characteristic space of a matrix is the set of all eigenvectors with the same eigenvalue, together with the zero vector. An eigenbasis for is any basis for the set of all vectors that consists of linearly independent eigenvectors of . Not every matrix has an eigenbasis, but every symmetric matrix does.The prefix eigen- is adopted from the German word eigen for "own-", "unique to", "peculiar to", or "belonging to" in the sense of "idiosyncratic" in relation to the originating matrix.Eigenvalues and eigenvectors have many applications in both pure and applied mathematics. They are used in matrix factorization, in quantum mechanics, and in many other areas.