Eigenvector Examples Definition Identifying Eigenvectors

In applied mathematics and physics the eigenvectors of a matrix or a differential operator often have important physical significance. In classical mechanics the eigenvectors of the governing equations typically correspond to natural modes of vibration in a body, and the eigenvalues to their frequencies. In quantum mechanics, operators correspond to observable variables, eigenvectors are also called eigenstates, and the eigenvalues of an operator represent those values of the corresponding variable that have non-zero probability of occurring.

1 Examples

Intuitively, for linear transformations of two-dimensional space R², eigenvectors are thus:

2 Definition

Formally, we define eigenvectors and eigenvalues as follows: If A : V -> V is a linear operator on some vector space V, v is a non-zero vector in V and c is a scalar (possibly zero) such that

then we say that v is an eigenvector of the operator A, and its associated eigenvalue is . Note that if v is an eigenvector with eigenvalue , then any non-zero multiple of v is also an eigenvector with eigenvalue . In fact, all the eigenvectors with associated eigenvalue , together with 0, form a subspace of V, the eigenspace for the eigenvalue .

3 Identifying eigenvectors

and therefore 2 is an eigenvalue of A and we have found a corresponding eigenvector.

4 The characteristic polynomial

An important tool for describing eigenvalues of square matrices is the characteristic polynomial: saying that c is an eigenvalue of A is equivalent to stating that the system of linear equations (A - cI) x = 0 (where I is the identity matrix) has a non-zero solution x (namely an eigenvector), and so it is equivalent to the determinant det(A - c I) being zero. The function p(c) = det(A - cI) is a polynomial in c since determinants are defined as sums of products. This is the characteristic polynomial of A; its zeros are precisely the eigenvalues of A. If A is an n-by-n matrix, then its characteristic polynomial has degree n and A can therefore have at most n eigenvalues.

Returning to the example above, if we wanted to compute all of A's eigenvalues, we could determine the characteristic polynomial first:

and because we see that the eigenvalues of A are 2, 1 and -1. The Cayley-Hamilton theorem states that every square matrix satisfies its own characteristic polynomial.

(In practice, eigenvalues of large matrices are not computed using the characteristic polynomial. Faster and more numerically stable methods are available, for instance the QR decompositionIn linear algebra, the QR decomposition of a matrix A is a factorization expressing A as A QR where Q is an orthogonal matrix QQ''T I , and R is an upper triangular matrix. The QR decomposition is often used to solve the linear least squares problem..)

5 Complex eigenvectors

Note that if A is a realIn mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. The term "real number" is a retronym coined in response to " imaginary number". Real numbers may matrix, the characteristic polynomial will have real coefficients, but not all its roots will necessarily be real. The complexThe complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. The complex numbers contain a number , the imaginary unit with , i. is a square root of. Every complex number can be represented in the form , wher eigenvalues will all be associated to complex eigenvectors.

In general, if v₁, ..., v_m are eigenvectors to different eigenvalues λ₁, ..., λ_m, then the vectors v₁, ..., v_m are necessarily linearly independent.

The spectral theoremIn mathematics, particularly linear algebra and functional analysis, the spectral theorem is a collection of results about linear operators or about matrices. In broad terms the spectral theorem provides conditions under which an operator or a matrix can for symmetric matrices states that, if A is a real symmetric n-by-n matrix, then all its eigenvalues are real, and there exist n linearly independent eigenvectors for A which all have length 1 and are mutually orthogonal.

Our example matrix from above is symmetric, and three mutually orthogonal eigenvectors of A are

These three vectors form a basis of R³. With respect to this basis, the linear map represented by A takes a particularly simple form: every vector x in R³ can be written uniquely as

and then we have

6 Infinite-dimensional spaces

The concept of eigenvectors can be extended to linear operators acting on infinite-dimensional Hilbert spaces or

Banach spaces.

There are operators on Banach spaces which have no eigenvectors at all. For example, take the bilateral shift on the Hilbert space

it is easy to see that any potential

eigenvector can't be square-summable, so none exist. However, any bounded linear operator on a Banach space V does have non-empty spectrum. The spectrum of the operator
T : V → V is defined as

is not invertible

Then is a compact set of complex numbers, and it is non-empty. When is a

compact operator (and in particular when is an

operator between finite-dimensional spaces as above), the spectrum of is the same as the set of its eigenvalues.

The spectrum of an operator is an important property in functional analysis.

7 See also

8 External links

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Vectors \| Vector spaces \| Linear span \| Linear transformation \| Linear independence \| Linear combination \| Basis \| Column space \| Row space \| Dual space \| Orthogonality \| Eigenvector \| Eigenvalue \| Least squares regressions \| Outer product \| Cross product \| Dot product \| Transpose \| Matrix decomposition

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