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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. This vector is then called the eigenvector associated with the eigenvalue. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ...
For the square matrix section, see square matrix. ...
The word vector means carrier in Latin; it is derived from the Latin verb vehere, which means to carry. ...
In linear algebra, the eigenvectors (from the German eigen meaning inherent, characteristic) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ...
The eigenvalues of a matrix or a differential operator often have important physical significance. In classical mechanics the eigenvalues of the governing equations typically correspond to the natural frequencies of vibration (see resonance). In quantum mechanics, the eigenvalues of an operator corresponding to some observable variable are those values of the observable that have non-zero probability of occurring. 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. ...
This article is about resonance in physics. ...
Fig. ...
The word eigenvalue comes from the German Eigenwert which means "proper or characteristic value."
Definition Formally, we define eigenvectors and eigenvalues as follows. Let A be an n-by-n matrix of real number or complex numbers (see below for generalizations). We say that λ ∈ C is an eigenvalue of A with eigenvector v ∈ Cn if v is not zero and The word real has many different meanings: Real is something that exists, in the physical sense. ...
In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit , satisfying . ...
- Av = λv.
The spectrum of A, denoted σ(A), is the set of all eigenvalues.
Computing eigenvalues Suppose that we want to compute the eigenvalues of a given matrix. If the matrix is small, we can compute them symbolically using the characteristic polynomial. However, this is often impossible for larger matrices, in which case we must use a numerical method.
Symbolic computations using the characteristic polynomial The eigenvalues of a matrix are the zeros of its characteristic polynomial. Indeed, if λ is an eigenvalue of A with eigenvector v, then (A − λI)v = 0, where I denotes the identity matrix. This is only possible if the determinant of A − λI vanishes. But the characteristic polynomial is defined to be pA(λ) = det(A − λI). In linear algebra, one associates a polynomial to every square matrix, its characteristic polynomial. ...
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. ...
It follows that we can compute all the eigenvalues of a matrix A by solving the equation pA(λ) = 0. The fundamental theorem of algebra says that this equation has at least one solution, so every matrix has at least one eigenvalue. In mathematics, the fundamental theorem of algebra states that every complex polynomial of degree n has exactly n zeroes, counted with multiplicity. ...
Numerical computations Main article: eigenvalue algorithm. In linear algebra, one of the most important problems is designing efficient and stable algorithms for finding the eigenvalues of a matrix. ...
The Abel-Ruffini theorem implies that there is no general algorithm for finding the zeros of the characteristic polynomial. Therefore, general eigenvalues algorithms are iterative. The easiest method is power iteration: we choose a random vector v and compute Av, A2v, A3v, ... This sequence will almost always converge to an eigenvector corresponding to the dominant eigenvalue. This algorithm is easy, but not very useful by itself. However, popular methods such as the QR algorithm are based on it. The Abel-Ruffini theorem states that there is no general solution in radicals to polynomial equations of degree five or higher. ...
An iterative method attempts to solve a problem (for example an equation or system of equations) by finding successive approximations to the solution starting from an initial guess. ...
In ordinary language, the word random is used to express apparent lack of purpose or cause. ...
A QR algorithm is a procedure to calculate the eigenvalues of a matrix. ...
Example Let us determine the eigenvalues of the matrix We first compute the characteristic polynomial of A: -
This polynomial factorizes as p(λ) = − (λ − 2)(λ − 1)(λ + 1). Therefore, the eigenvalues of A are 2, 1 and −1.
Multiplicity The (algebraic) multiplicity of an eigenvalue λ of A is the order of λ as a zero of the characteristic polynomial of A; in other words, it is the number of factors t − λ in the characteristic polynomial. An n-by-n matrix has n eigenvalues, counted according to their algebraic multiplicity, because its characteristic polynomial has degree n.
An eigenvalue of algebraic multiplicity 1 is called a simple eigenvalue.
Occasionally, in an article on matrix theory, one may read a statement like
- "the eigenvalues of a matrix A are 4,4,3,3,3,2,2,1,"
meaning that the algebraic multiplicity of 4 is two, of 3 is three, of 2 is two and of 1 is one. This style is used because algebraic multiplicity is the key to many mathematical proofs in matrix theory.
The geometric multiplicity of an eigenvalue λ is the dimension of the associated eigenspace, which consists of all the eigenvectors associated with λ; in other words, it is the nullity of the matrix λI − A. The geometric multiplicity is less than or equal to the algebraic multiplicity. In linear algebra, the nullity of a matrix M is the number of columns of M minus the rank of M. If the m by n matrix M is regarded as a linear transformation Rn → Rm, then the nullity is equal to the dimension of the kernel of this linear...
Consider for example the matrix
It has only one eigenvalue, namely λ = 1. The characteristic polynomial is (λ − 1)2, so this eigenvalue has algebraic multiplicity 2. However, the associated eigenspace is spanned by (1, 0)T, so the geometric multiplicity is only 1.
Properties The spectrum is invariant under similarity transformations: the matrices A and P-1AP have the same eigenvalues for any matrix A and any invertible matrix P. The spectrum is also invariant under transposition: the matrices A and AT have the same eigenvalues. Several equivalence relations in mathematics are called similarity. ...
In mathematics and especially linear algebra, an n-by-n matrix A is called invertible, non-singular or regular if there exists another n-by-n matrix B such that AB = BA = In, where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ...
See transposition for meanings of this term in telecommunication and music. ...
A matrix is invertible if and only if zero is not an eigenvalue of the matrix.
A matrix is diagonalizable if and only if the algebraic and geometric multiplicities coincide for all its eigenvalues. In particular, an n-by-n matrix is diagonalizable if it has n different eigenvalues. In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i. ...
The location of the spectrum is often restricted if the matrix has a special form:
Generally, the trace of a matrix equals the sum of the eigenvalues, and the determinant equals the product of the eigenvalues (counted according to algebraic multiplicity). A Hermitian matrix (or self-adjoint matrix) is a square matrix with complex entries so that the matrix is equal to its own conjugate transpose - that is, if the element in the ith row and jth column is equal to the complex conjugate of the element in the jth row...
In linear algebra, the positive-definite matrices are (in several ways) analogous to the positive real numbers. ...
In linear algebra, a square matrix (or more generally, a linear transformation from a complex vector space with a sesquilinear norm to itself) A is said to be skew-Hermitian or antihermitian if its conjugate transpose A* is also its negative. ...
In mathematics, a unitary matrix is a n by n complex matrix U satisfying the condition where In is the identity matrix and U* is the conjugate transpose (also called the Hermitian adjoint) of U. Note this condition says that a matrix U is unitary if it has an inverse...
The graph of the absolute value function In mathematics, the absolute value (or modulus) of a real number is its numerical value without regard to its sign. ...
In the mathematical discipline of linear algebra, a triangular matrix is a special kind of square matrix where the entries below or above the main diagonal are zero. ...
In linear algebra, a diagonal matrix is a square matrix in which only the entries in the main diagonal are non-zero. ...
In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i. ...
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. ...
Suppose that A is an m-by-n matrix, with m ≤ n, and that B is an n-by-m matrix. Then BA has the same eigenvalues as AB plus m − n eigenvalues equal to zero.
Extensions and generalizations
Eigenvalues of an operator Suppose we have a linear operator A mapping the vector space V to itself. As in the matrix case, we say that λ ∈ C is an eigenvalue of A if there exists a nonzero v ∈ V such that Av = λv. A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ...
Suppose now that A is a bounded linear operator on a Banach space V. We say that λ ∈ C is a spectral value of A if the operator A − λI is not invertible, where I denotes the identity operator. Note that by the closed graph theorem, if a bounded operator has an inverse, the inverse is necessarily bounded. The set of all spectral values is the spectrum of A. In mathematics, the operator norm is a norm defined on the space of bounded operators between two Banach spaces. ...
In mathematics, Banach spaces, named after Stefan Banach who studied them, are one of the central objects of study in functional analysis. ...
In mathematics, the closed graph theorem is a basic result of functional analysis. ...
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. ...
If V is finite dimensional, then the spectrum of A is the same of the set of eigenvalues of A. This follows from the fact that on finite-dimensional spaces injectivity of a linear operator A is equivalent to surjectivity of A. However, an operator on an infinite-dimensional space may have no eigenvalues at all, while it always has spectral values. In mathematics, an injective function (or one-to-one function or injection) is a function which maps distinct input values to distinct output values. ...
In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ...
Eigenvalues of a matrix with entries from a ring Suppose that A is a square matrix with entries in a ring R. An element λ ∈ R is called a right eigenvalue of A if there exists a nonzero column vector x such that Ax=λx, or a left eigenvalue if there exists a nonzero row vector y such that yA=yλ. In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers. ...
If R is commutative, the left eigenvalues of A are exactly the right eigenvalues of A and are just called eigenvalues. If R is not commutative, e.g. quaternions, they may be different. In mathematics, the quaternions are a non-commutative extension of the complex numbers. ...
Eigenvalues of a graph An eigenvalue of a graph is defined as an eigenvalue of the graph's adjacency matrix A, or (increasingly) of the graph's Laplacian matrix I − T − 1 / 2AT − 1 / 2, where T is a diagonal matrix holding the degree of each vertex, and in T − 1 / 2, 0 is substituted for 0 − 1 / 2. In mathematics and computer science, the adjacency matrix for a finite graph on n vertices is an n × n matrix in which entry aij is the number of edges from vi to vj in . ...
Generalized eigenvalue problem A generalized eigenvalue problem is of the form where A and B are matrices (with complex entries). The generalized eigenvalues λ can be obtained by solving the equation If B is invertible, then problem (1) can be obviously written in the form which is a standard eigenvalue problem. However, in most situations it is preferable not to perform the inversion, and solve the generalized eigenvalue problem as stated originally.
If A and B are symmetric matrices with real entries, then problem (1) has real eigenvalues. This would have not been easy to see from the equivalent formulation (2), because the matrix B − 1A is not necessarily symmetric if A and B are.
External links - Eigenvalue (of a matrix) (http://planetmath.org/?op=getobj&from=objects&id=4397) on PlanetMath.
Wikibooks Algebra has more about this subject: Eigenvalues and eigenvectors PlanetMath is a free, collaborative, online mathematics encyclopedia. ...
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