Eigenvalues and Eigenvectors

Eigenvalues And Eigenvectors

An eigenvector of a square matrix is a non-zero vector that, when multiplied by the matrix, yields a vector that differs from the original at most by a multiplicative scalar.

For example, if three-element vectors are seen as arrows in three-dimensional space, an eigenvector of a 3×3 matrix A is an arrow whose direction is either preserved or exactly reversed after multiplication by A. The corresponding eigenvalue determines how the length and sense of the arrow is changed by the operation.

Specifically, a non-zero column vector v is a right eigenvector of a matrix A if (and only if) there exists a number λ such that Av = λv. If the vector satisfies vA = λv instead, it is said to be a left eigenvector. The number λ is called the eigenvalue corresponding to that vector. The set of all eigenvectors of a matrix, each paired with its corresponding eigenvalue, is called the eigensystem of that matrix.

An eigenspace of A is the set of all eigenvectors with the same eigenvalue, together with the zero vector.

The terms characteristic vector, characteristic value, and characteristic space are also used for these concepts. The prefix eigen- is adopted from the German word eigen for "self". Having an eigenvalue is an accidental property of a real matrix (since it may fail to have an eigenvalue), but every complex matrix has an eigenvalue.

These concepts are naturally extended to more general situations, where the set of real scale 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 concept of "parallel to" is interpreted as "scalar multiple of", and the "vector" in "eigenvector" may be replaced by a more specific term, as in "eigenfunction", "eigenmode", "eigenface", "eigenstate", and "eigenfrequency". Thus, for example, the exponential function is an eigenfunction of the derivative operator " ", with eigenvalue, since its derivative is .

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.