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In linear algebra, the eigenvectors (from the German eigen meaning "own") of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. The scalar is then called the eigenvalue associated with the eigenvector. Download high resolution version (640x625, 61 KB) From: http://data2. ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... The word vector means carrier in Latin; it is derived from the Latin verb vehere, which means to carry. ... Scalar is a concept that has meaning in mathematics, physics, and computing. ... In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ...

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 being measured. Applied mathematics is a branch of mathematics that concerns itself with the application of mathematical knowledge to other domains. ... The willingness to question previously held truths and search for new answers resulted in a period of major scientific advancements, now known as the Scientific Revolution. ... For the square matrix section, see square matrix. ... In mathematics, a differential operator is a linear operator defined as a function of the differentiation operator. ... In physics, Classical mechanics is one of the two major sub-fields of study in the science of mechanics, which is concerned with the motions of bodies, and the forces that cause them. ... Fig. ... In physics, particularly in quantum physics a system observable is a property of the system state that can be determined by some sequence of physical operations. ... The word probability derives from the Latin probare (to prove, or to test). ...

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 λ is a scalar (possibly zero) such that A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ...

then we say that v is an eigenvector of the operator A, and its associated eigenvalue is λ. Note that if v is an eigenvector with eigenvalue λ, then any non-zero multiple of v is also an eigenvector with eigenvalue λ. In fact, all the eigenvectors with associated eigenvalue λ, together with 0, form a subspace of V, the eigenspace for the eigenvalue λ. In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ...

Examples

For linear transformations of two-dimensional space R² we can discern the following special cases: In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... Dimension (from Latin measured out) is, in essence, the number of degrees of freedom available for movement in a space. ...

• translations: no eigenvectors. Example: Av=v+a
• rotations: no real eigenvectors, (Complex eigenvalue, eigenvector pairs exist). Example: A=(01)(10).
• reflection: eigenvectors are perpendicular and parallel to the line of symmetry, the eigenvalues are -1 and 1, respectively. Example: A=(10)(0 -1)
• uniform scaling: all vectors are eigenvectors, and the eigenvalue is the scale factor.Example: Ax=cx
• projection onto a line: vectors on the line are eigenvectors with eigenvalue 1 and vectors perpendicular to the line are eigenvectors with eigenvalue 0. Example: A=(00)(01).

Translation is an activity comprising the interpretation of the meaning of a text in one language — the source text — and the production of a new, equivalent text in another language — called the target text, or the translation. ... Rotation is the movement of a body in such a way that the distance between a certain fixed point and any given point of that body remains constant. ... The word reflection (also spelt reflexion in British English) can refer to several different concepts: In mathematics, reflection is the transformation of a space. ... Square with symmetry group D4 Symmetry is a characteristic of geometrical shapes, equations, and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ... A scale factor is a number which scales some quantity. ...

Identifying eigenvectors

For example, consider the matrix For the square matrix section, see square matrix. ...

which represents a linear operator R³ -> R³. One can check that

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

The characteristic polynomial

An important tool for describing eigenvalues of square matrices is the characteristic polynomial: saying that λ is an eigenvalue of A is equivalent to stating that the system of linear equations (A - λid_V) v = 0 (where id_V is the identity matrix) has a non-zero solution v (namely an eigenvector), and so it is equivalent to the determinant det(A - λ id_V) being zero. The function p(λ) = det(A - λid_V) is a polynomial in λ 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. In linear algebra, one associates a polynomial to every square matrix, its characteristic polynomial. ... In mathematics and linear algebra, a system of linear equations is a set of linear equations such as 3x1 + 2x2 − x3 = 1 2x1 − 2x2 + 4x3 = −2 −x1 + ½x2 − x3 = 0. ... In linear algebra, the identity matrix of size n is the n-by-n square matrix with ones on the main diagonal and zeros elsewhere. ... In linear algebra, the determinant is a function that associates a scalar det(A) to every square matrix A. The fundamental geometric meaning of the determinant is as the scale factor for volume when A is regarded as a linear transformation. ... In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ...

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

and 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, that is p(A)=0. In linear algebra, the Cayley-Hamilton theorem (named after the mathematicians Arthur Cayley and William Hamilton) states that every square matrix over the real or complex field, satisfies its own characteristic equation. ...

(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 decomposition.) In linear algebra, the QR decomposition of a matrix is a factorization expressing as where is an orthogonal matrix (), and is an upper triangular matrix. ...

Complex eigenvectors

Note that if A is a real matrix, the characteristic polynomial will have real coefficients, but not all its roots will necessarily be real. The complex eigenvalues will all be associated to complex eigenvectors. In 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. ... In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ...

In general, if v₁, ..., v_m are eigenvectors with different eigenvalues λ₁, ..., λ_m, then the vectors v₁, ..., v_m are necessarily linearly independent. In linear algebra, a set of elements of a vector space is linearly independent if none of the vectors in the set can be written as a linear combination of finitely many other vectors in the set. ...

The spectral theorem 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 are mutually orthogonal. In mathematics, particularly linear algebra and functional analysis, the spectral theorem is a collection of results about linear operators or about matrices. ... In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ...

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 In mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions: B is both a set of linearly independent vectors and a generating set of V. B is a minimal generating set of V... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are...

x = x₁v₁ + x₂v₂ + x₃v₃

and then we have

Ax = 2x₁v₁ + x₂v₂ − x₃v₃.

Decomposition theorem

An n by n matrix has n linearly independent real eigenvectors if and only if it can be decomposed into the form

and Λ is a diagonal matrix with all of the eigenvalues on the diagonal. If A is symmetric, then U is orthogonal, and if A is Hermitian, then U is unitary. Such a matrix U does not always exist; for example In linear algebra, a diagonal matrix is a square matrix in which only the entries in the main diagonal are non-zero. ... Symmetry is a characteristic of geometrical shapes, equations and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ... In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ... A number of mathematical entities are named Hermitian, after the mathematician Charles Hermite: Hermitian matrix Hermitian operator Hermitian adjoint Hermitian form Hermitian metric See also: self-adjoint This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... In government, see Unitary state In mathematics, see Unitary matrix Unitary operator Unitary group Unitary representation This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ...

has only one 1-dimensional eigenspace. In such a case, the singular value decomposition must be used. In linear algebra the singular value decomposition (SVD) is an important factorization of a rectangular real or complex matrix, with several applications in signal processing and statistics. ...

Infinite-dimensional spaces

The concept of eigenvectors can be extended to linear operators acting on infinite-dimensional Hilbert spaces or Banach spaces. In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... In mathematics, a Hilbert space is an inner product space that is complete with respect to the norm defined by the inner product. ... In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ...

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 σ(A) of the operator A : V → V is defined as In mathematics, and in particular functional analysis, the shift operators are examples of linear operators, important for their simplicity and natural occurrence. ...

Then σ(A) is a compact set of complex numbers, and it is non-empty. When A is a compact operator (and in particular when A is an operator between finite-dimensional spaces as above), the spectrum of A is the same as the set of its eigenvalues. In mathematics, a compact set is a set of points in a topological space such that every one of its (possibly infinite) open covers has a finite subcover. ... In functional analysis, a compact operator is a linear operator L from a Banach space X to another Banach space Y, such that the image under L of any bounded subset of X is a relatively compact subset of Y. Such an operator is necessarily a bounded operator, and so... In mathematics, a set is called finite if and only if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ...

The spectrum of an operator is an important property in functional analysis. Functional analysis is that branch of mathematics and specifically of analysis which is concerned with the study of spaces of functions. ...

See also

• eigenplane
• spectrum of an operator
• spectral theorem

In mathematics, an eigenplane is a two-dimensional invariant subspace in a given vector space. ... In functional analysis, the concept of the spectrum of an operator is a generalisation of the concept of eigenvalues, which is much more useful in the case of operators on infinite-dimensional spaces. ... In mathematics, particularly linear algebra and functional analysis, the spectral theorem is a collection of results about linear operators or about matrices. ...

External links

• MathWorld: Eigenvector
• Earliest Known Uses of Some of the Words of Mathematics: E - see eigenvector and related terms