In this transformation of the Mona Lisa, the blue vector has been rotated, but the red one has not. The red vector is thus an eigenvector of the transformation. Since it has been neither stretched nor compressed, its eigenvalue is equal to one. All vectors parallel to it are also eigenvectors, with the same eigenvalue. They form the eigenspace associated with this eigenvalue.

In mathematics, and in particular in vectorial analysis, the eigenvalues, eigenvectors, and eigenspaces of a transformation (from a vector space to itself) are important properties of this transformation. These key concepts play a major role in mathematics and applied disciplines. Image File history File links Eigen. ... Image File history File links Eigen. ... The Mona Lisa (Italian, Spanish: La Gioconda; French: La Joconde), is an oil painting on poplar wood by the famous Italian Renaissance artist Leonardo da Vinci. ... Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... In mathematics, a transformation in elementary terms is any of a variety of different operations from geometry, such as rotations, reflections and translations. ... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ...

The prefix eigen emphasizes the fact that these properties are important characteristics of the transformation. In many common cases knowing all eigenvalues and eigenvectors of a transformation is equivalent to the explicit knowledge of the transformation. The word eigen is German for "own", "peculiar", or "individual": the most likely translation into English mathematical jargon would be "characteristic", and some older references do use the expressions "characteristic value", "characteristic vector" and so forth, or even "eigenwert" which is German for eigenvalue, but the more distinctive term "eigenvalue" has now become standard.

Definitions

Transformations of space—such as translation, rotation, reflection, stretching, compression, or any combination of these (other, possibly nonlinear, transformations could also be listed)—may be visualized by the effect they produce on vectors (one-dimensional arrows pointing from one point to another). In mathematics, a transformation in elementary terms is any of a variety of different operations from geometry, such as rotations, reflections and translations. ... 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. ... Stretching is the activity of gradually applying tensile force to lengthen, strengthen, and lubricate muscles, often performed in anticipation of physical exertion and to increase the range of motion within a joint. ... There are several types of compression: physical compression data compression multimedia compression image compression executable compression audio compression video compression bandwidth compression audio level compression compression (functional analysis) See also Arch bridge Compression arch suspended-deck bridge Compressor Compression ratio This is a disambiguation page — a list of pages that... To do: 20th century mathematics chaos theory, fractals Lyapunov stability and non-linear control systems non-linear video editing See also: Aleksandr Mikhailovich Lyapunov Dynamical system External links http://www. ... In physics and engineering, a vector is a physical entity which has a magnitude which is a scalar (a physical quantity expressed as the product of a numerical value and a physical unit, not just a number). ... A spatial point is an entity with a location in space but no extent (volume, area or length). ...

• Eigenvectors of transformations are vectors^[2] which are either left unaffected or merely multiplied by a scale factor after the transformation.
• An eigenvector's eigenvalue is the scale factor that it has been multiplied by.
• An eigenspace is a space consisting of all eigenvectors which have the same eigenvalue.
• The geometric multiplicity of an eigenvalue is the dimension of the associated eigenspace.
• The spectrum of a transformation is the set of all its eigenvalues.

For instance, an eigenvector of a rotation is a vector located within the axis around which the rotation is performed. The corresponding eigenvalue' is 1 and the corresponding eigenspace contains all the vectors parallel to the axis. As this is a one-dimensional space, its geometric multiplicity is one. This is the only eigenvalue of the spectrum (of this rotation) that is a real number. A scale factor is a number which scales some quantity. ... This article is about space — the scientific and philosophical concepts. ... The axis of rotation of a rotating body is a line such that the distance between any point on the line and any point of the body remains constant under the rotation. ... Parallel is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or planes, or a combination of these. ... 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. ...

See also: eigenplane In mathematics, an eigenplane is a two-dimensional invariant subspace in a given vector space. ...

Examples

As the Earth rotates, every arrow pointing outward from the center of the Earth also rotates, except those arrows that lie on the axis of rotation. Consider the transformation of the Earth after one hour of rotation. An arrow from the center of the Earth to the South Pole would be an example of an eigenvector of this transformation, while an arrow from the center of the Earth to Paris would not be an eigenvector. Since the arrow pointing at the pole is not stretched by the rotation of the Earth, its eigenvalue would be 1. Location of the South Pole in the Antarctic continent. ... The Eiffel Tower has become a symbol of Paris throughout the world. ...

As another example, consider a thin metal sheet expanding uniformly about a fixed point. The transformation that expands every point on the sheet to twice its original distance from the fixed point has eigenvalue 2. Every vector from the fixed point to a point on the sheet is an eigenvector, and the eigenspace is the set of all these vectors.

A standing wave in a rope fixed at its boundaries can be seen as an example of an eigenvector (more precisely, an eigenfunction) of the transformation corresponding to the passage of time. As time passes, the standing wave is scaled but its shape is not modified. In this case the eigenvalue is time dependent.

However, three-dimensional geometric space is not the only vector space. For example, consider a stressed rope fixed at both ends, like the vibrating strings of a string instrument. The distances of atoms of the vibrating rope from their positions when the rope is at rest can be seen as the components of a vector in a space with as many dimensions as atoms in the rope. Animation of a standing wave. ... Animation of a standing wave. ... A wave is a disturbance that propagates, carrying energy. ... A standing wave, also known as a stationary wave, is a wave that remains in a constant position. ... A vibration in a string is a wave. ... A string instrument (also stringed instrument) is a musical instrument that produces sound by means of vibrating strings. ... In general, a things components are its parts; the things that compose it. ... Properties For alternative meanings see atom (disambiguation). ...

If one considers the transformation of the rope as time passes, its eigenvectors (or eigenfunctions, if one assumes the rope is a continuous medium) are its standing waves—the things that, mediated by the surrounding air, humans can experience as the twang of a bowstring or the plink of a guitar. The standing waves correspond to particular oscillations of the rope such that the shape of the rope is scaled by a factor (the eigenvalue) as time passes. Each component of the vector associated with the rope is multiplied by this time-dependent factor. Taking the damping of the oscillations into account, the amplitude of the standing waves (their eigenvalues) decrease with time. One can then associate a lifetime with the eigenvector, and relate the concept of an eigenvector to the concept of resonance (a key concept in physics).-1... A bow is a weapon that shoots arrows powered by the elasticity of the bow and/or its string. ... The classical guitar typically has 3 nylon and 3 nickel-wound strings. ... Damping is any effect, either deliberately engendered or inherent to a system, that tends to reduce the amplitude of oscillations. ... Amplitude is a nonnegative scalar measure of a waves magnitude of oscillation, that is, magnitude of the maximum disturbance in the medium during one wave cycle. ... Lifetime can refer to: The length of time a person is alive. ... In physics, resonance is the tendency of a system to absorb more oscillatory energy when the frequency of the oscillations matches the systems natural frequency of vibration (its resonant frequency) than it does at other frequencies. ...

Eigenvalue equation

Mathematically, v_λ is an eigenvector and λ the corresponding eigenvalue of a transformation if the equation In mathematics, one often (not quite always) distinguishes between an identity, which is an assertion that two expressions are equal regardless of the values of any variables that occur within them, and an equation, which may be true for only some (or none) of the values of any such variables. ...

is true, where is the vector obtained when applying the transformation to v_λ.

Suppose is a linear transformation (which means, if for all scalars a, b, and vectors v, w), and that the vector space has a basis (finite-dimensional vector spaces always do). Then can be represented (relative to that basis) by the matrix (or two-dimensional array) T and v_λ by the one-dimensional vertical array v_λ. The eigenvalue equation in its matrix representation 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. ... Scalar is a concept that has meaning in mathematics, physics, and computing. ... 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 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. ... For the square matrix section, see square matrix. ... In computer programming, an array, also known as a vector or list, is one of the simplest data structures. ... In computer programming, an array, also known as a vector or list, is one of the simplest data structures. ... For the square matrix section, see square matrix. ...

where the juxtaposition is matrix multiplication. This is equivalent to a set of n linear equations (where n is the number of basis vectors in the basis set). In this equation both the eigenvalue λ and the n components of v_λ are unknowns. This article gives an overview of the various ways to multiply matrices. ... 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... The term Unknown is used in a number of contexts: To indicate the lack of knowledge To refer to anonymity In mathematics, as a noun, the unknowns of an equation are the quantities (or variables) which are solutions of it Unknown is the title of a fantasy magazine published from...

Sometimes, when is a nonlinear transformation, or when it is not possible or difficult to provide a basis set (this will usually mean the dimension of the vector space is very large or infinite; for example, the case of the rope above), the eigenvalue equation cannot be written down in a matrix form. In this case it can be advantageous to represent the eigenvalue equation as a set of nonlinear equations or of differential equations depending on the nature of the transformation and the space to which it applies. To do: 20th century mathematics chaos theory, fractals Lyapunov stability and non-linear control systems non-linear video editing See also: Aleksandr Mikhailovich Lyapunov Dynamical system External links http://www. ... In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ...

If is a differential operator, the eigenvectors are commonly called eigenfunctions of the differential operator representing . For example, differentiation itself is a linear transformation since (if M and N are differentiable functions, and a and b are constants) In mathematics, a differential operator is a linear operator defined as a function of the differentiation operator. ... Differentiation can mean the following: In biology: cellular differentiation; evolutionary differentiation; In mathematics: see: derivative In cosmogony: planetary differentiation Differentiation (geology); Differentiation (logic); Differentiation (marketing). ... In mathematics, the derivative of a function is one of the two central concepts of calculus. ... In mathematics and the mathematical sciences, a constant is a fixed, but possibly unspecified, value. ...

d(aM + bN) / dt = adM / dt + bdN / dt

Consider differentiation with respect to time, t. Its eigenfunctions obey the eigenvalue equation

dN / dt = λN,

where λ is the eigenvalue associated with the function. Such a function of time is constant, if λ = 0; grows proportionally to itself, if λ is positive; and decays proportionally to itself, if λ is negative. For example, an (idealized) population of rabbits breeds faster the more rabbits there are, and thus satisfies the equation with a positive lambda.

The solution to the eigenvalue equation is N = exp(λt), the exponential function; thus that function is an eigenfunction of the differential operator d/dt with the eigenvalue λ. If λ is negative, we call the evolution of N an exponential decay; if it is positive, an exponential growth. The value of λ can be any complex number. The spectrum of d/dt is therefore the whole complex plane. In this example the vector space in which the operator d/dt acts is the space of the differentiable functions of one variable. This space has an infinite dimension (because it is not possible to express any differentiable function as a linear combination of a finite number of basis functions). However, the eigenspace associated with any given eigenvalue λ is one dimensional. It is the set of all functions N = N₀exp(λt) . N₀ is an arbitrary constant, the initial population at t=0. The exponential function is one of the most important functions in mathematics. ... Negative has meaning in several contexts: Look up negative in Wiktionary, the free dictionary Negative and non-negative numbers Negative (photography) In optics, diverging lenses are also called negative lenses. ... A quantity is said to be subject to exponential decay if it decreases at a rate proportional to its value. ... In common usage positive is sometimes used in affirmation, as a synonym for yes or to express certainty. Look up Positive on Wiktionary, the free dictionary In mathematics, a number is called positive if it is bigger than zero. ... In mathematics, a quantity that grows exponentially is one that grows at a rate proportional to its size. ... In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ... The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ... In mathematics, the derivative of a function is one of the two central concepts of calculus. ... In computer science and mathematics, a variable is a symbol denoting a quantity or symbolic representation. ... Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ... In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ... In functional analysis and its applications, a function space can be viewed as a vector space of infinite dimension whose basis vectors are functions not vectors. ...

Spectral theorem

Main article: spectral theorem

The spectral theorem shows the whole importance of the eigenvalues and eigenvectors for characterizing a linear transformation in a unique way. In its simplest version, the spectral theorem states that, under precise conditions (see Spectral theorem), a linear transformation of a vector can be expressed as the linear combination of the eigenvectors with coefficients equal to the eigenvalues times the scalar product (or dot product) of the eigenvectors with the vector on which the transformation is applied. Mathematically, it can be written In mathematics, particularly linear algebra and functional analysis, the spectral theorem is a collection of results about linear operators or about matrices. ... In mathematics, particularly linear algebra and functional analysis, the spectral theorem is a collection of results about linear operators or about matrices. ... In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ... In mathematics, a coefficient is a multiplicative factor that belongs to a certain object such as a variable (for example, the coefficients of a polynomial), a basis vector, a basis function and so on. ... In mathematics, the dot product (also known as the scalar product and the inner product) is a function (·) : V × V → F, where V is a vector space and F its underlying field. ... In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors and returns a scalar quantity. ...

where and stand for the eigenvectors and eigenvalues of . The simplest case in which the theorem is valid is the case where the linear transformation is given by a real symmetric matrix or complex Hermitian matrix. 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 linear algebra, a symmetric matrix is a matrix that is its own transpose. ... In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ... A Hermitian matrix (or self-adjoint matrix) is a square matrix with complex entries which is equal to its own conjugate transpose - that is, the element in the ith row and jth column is equal to the complex conjugate of the element in the jth row and ith column, for...

If one defines the qth power of transformations as q successive transformations, one can also define polynomials of transformations. A more general version of the theorem is that any polynomial P of is equal to In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ...

The theorem can be extended to other functions of transformations like analytic functions, the most general case being Borel functions. In mathematics, an analytic function is a function that is locally given by a convergent power series. ... In mathematics, measurable functions are well-behaved functions between measurable spaces. ...

Eigenvalues of matrices

Computing eigenvalues of matrices

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. In linear algebra, one associates a polynomial to every square matrix, its characteristic polynomial. ... Numerical analysis is the study of algorithms for the problems of continuous mathematics (as distinguished from discrete mathematics). ...

Symbolic computations using the characteristic polynomial

Main article: symbolic computation of matrix eigenvalues

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 - λI) v = 0 (where I is the identity matrix) has a non-zero solution v (namely an eigenvector), and so it is equivalent to the determinant det(A - λI) being zero. The function p(λ) = det(A - λI) is a polynomial in λ since determinants are defined as sums of products. This is the characteristic polynomial of A: the eigenvalues of a matrix are the zeros of its characteristic polynomial. It has been suggested that this article or section be merged with Eigenvalue algorithm. ... 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, a determinant is a function depending on n that associates a scalar det(A) to every n×n square matrix A. The fundamental geometric meaning of a 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. ... In linear algebra, one associates a polynomial to every square matrix, its characteristic polynomial. ...

It follows that we can compute all the eigenvalues of a matrix A by solving the equation p_A(λ) = 0. If A is an n-by-n matrix, then p_A has degree n and A can therefore have at most n eigenvalues. Conversely, the fundamental theorem of algebra says that this equation has exactly n roots (zeroes), counted with multiplicity. All real polynomials of odd degree have a real number as a root, so for odd n, every real matrix has at least one real eigenvalue. In the case of a real matrix, for even and odd n, the non-real eigenvalues come in conjugate pairs. In mathematics, the fundamental theorem of algebra states that every complex polynomial of degree n has exactly n roots (zeroes), counted with multiplicity. ... In mathematics, a root (or a zero) of a function f is an element x in the domain of f such that f(x) = 0. ...

An example of a matrix with no real eigenvalues is the 90-degree clockwise rotation:

whose characteristic polynomial is λ² + 1 and so its eigenvalues are the pair of complex conjugates i, -i. The associated eigenvectors are also not real.

Numerical computations

Main article: eigenvalue algorithm

In practice, eigenvalues of large matrices are not computed using the characteristic polynomial. Computing the polynomial becomes expensive in itself, and exact (symbolic) roots of a high-degree polynomial can be difficult to compute and express (for example the Abel-Ruffini theorem implies that they cannot be expressed simply using nth roots. Effective numerical algorithms for approximating roots of polynomials exist, but small errors in the eigenvalues can lead to large errors in the eigenvectors. Therefore, general algorithms to find eigenvectors and eigenvalues, are iterative. The easiest method is the power method: we choose a random vector v and compute Av, A²v, A³v, ... This sequence will after normalization 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. 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 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. ... The power method is a iterative approximative method for calculating the eigenvectors of a matrix. ... In ordinary language, the word random is used to express apparent lack of purpose or cause. ... This is a page about mathematics. ... A QR algorithm is a procedure to calculate the eigenvalues of a matrix. ...

Properties

Algebraic 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, if t is one root of the polynomial, it is the number of factors (t − λ) in the characteristic polynomial after factorization. An n-by-n matrix has n eigenvalues, counted according to their algebraic multiplicity, because its characteristic polynomial has degree n. Order is the opposite of anarchy and chaos. ... In mathematics, a root (or a zero) of a function f is an element x in the domain of f such that f(x) = 0. ... In mathematics, factorization or factoring is the decomposition of an object (for example, a number, a polynomial, or a matrix) into a product of other objects, or factors, which when multiplied together give the original. ...

An eigenvalue of algebraic multiplicity 1 is called a simple eigenvalue.

Occasionally, in an article on matrix theory, one may read a statement like Matrix theory is a branch of mathematics which focuses on the study of matrices. ...

"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. In mathematics, a proof is a demonstration that, given certain axioms, some statement of interest is necessarily true. ...

The algebraic multiplicity can also be thought of as a dimension: it is the dimension of the associated generalized eigenspace (1st sense), which is the nullspace of the matrix (λI − A)^k for any sufficiently large k. That is, it is the space of generalized eigenvectors (1st sense), where a generalized eigenvector is any vector which eventually becomes 0 if λI − A is applied to it enough times successively. Obviously any eigenvector is a generalized eigenvector, and so each eigenspace is contained in the associated generalized eigenspace. This provides an easy proof that the geometric multiplicity is always less than or equal to the algebraic multiplicity. (Do not confuse this 1st sense with generalized eigenvalue problem, below.) In this transformation of the Mona Lisa, the blue vector has been rotated, but the red one has not. ...

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 the axis usually called the x axis, spanned by the unit vector , so the geometric multiplicity is only 1. In the mathematical subfield of linear algebra, the linear span of a set of vectors is the set of all linear combinations of the vectors. ...

Decomposition theorem

The decomposition theorem is a version of the spectral theorem (see above) in the particular case of matrices. This theorem is usually introduced in terms of coordinate transformation. If U is an invertible matrix, it can be seen as a transformation from one coordinate system to another. In this new system the coordinates of the vector are labeled v'. The latter are obtained from the coordinates v in the original coordinate system by the relation v' = Uv and, the other way around, we have v = U ^{− 1}v'. Applying successively v' = Uv, w' = Uw and U ^{− 1}U = I, to the relation Av = w defining the matrix multiplication provides A'v' = w' with A' = UAU ^{− 1}, the representation of A in the new basis. The columns of U being the components of the new basis vectors within the old basis set. The decomposition theorem states that, if one chooses as columns of U n linearly independent eigenvectors of A, the new matrix A' = UAU ^{− 1} is diagonal and its diagonal elements are the eigenvalues of A. If this is possible the matrix A is diagonalizable. An example of non-diagonalizable matrix is given by the matrix A above. In this transformation of the Mona Lisa, the blue vector has been rotated, but the red one has not. ... 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. ... This article gives an overview of the various ways to multiply matrices. ... 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. ... In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i. ... In this transformation of the Mona Lisa, the blue vector has been rotated, but the red one has not. ...

There are several generalizations of this decomposition which can cope with the non-diagonalizable case, suited for different purposes:

• the singular value decomposition, A = UΣV ^* where Σ is diagonal but U is not necessarily equal to V;
• the Jordan normal form, where A = UΛU ^{− 1} where Λ is not diagonal but of a simple form with non vanishing entries only along the diagonal and one element above;
• any matrix A can be written uniquely as A = S + N where S is diagonalizable, N is nilpotent (i.e., such that N^q=0 for some q), and S commutes with N (SN=NS).
• any invertible matrix A can be written uniquely as A = SJ where S is diagonalizable and J is unipotent (i.e., such that the characteristic polynomial is a power of (λ-1), and S commutes with J).

In linear algebra singular value decomposition (SVD) is an important factorization of a rectangular real or complex matrix, with several applications in signal processing and statistics. ... In linear algebra, the Jordan normal form, also called the Jordan canonical form, named in honor of the 19th and early 20th-century French mathematician Camille Jordan, answers the question, for a given square matrix M over a field K containing the eigenvalues of M, to what extent can M... In mathematics, a nilpotent matrix is a square matrix that is nilpotent. ... In mathematics, a unipotent element r of a ring R is one such that r − 1 is a nilpotent element, i. ...

Other theorems

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 A^T have the same eigenvalues. In mathematics, an invariant is something that does not change under a set of transformations. ... 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. ... In mathematics, and in particular linear algebra, the transpose of a matrix is another matrix, produced by turning rows into columns and vice versa. ...

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 which has n different eigenvalues is always diagonalizable. In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i. ...

The vector space on which the matrix acts is always the direct sum of the generalized eigenspaces (i.e., is spanned by them and they are independent). This is true of the ordinary (non-generalized) eigenspaces if and only if they are equal to the generalized eigenspaces, i.e., if and only if the matrix is diagonalizable.

The location of the spectrum is often restricted if the matrix has a special form:

• All eigenvalues of a Hermitian matrix (A = A^*) are real. Furthermore, all eigenvalues of a positive-definite matrix (v^*Av > 0 for all vectors v) are positive.
• All eigenvalues of a skew-Hermitian matrix (A = −A^*) are purely imaginary.
• All eigenvalues of a unitary matrix (A^-1 = A^*) have absolute value one.
• The eigenvalues of a triangular matrix are the entries on the main diagonal. This holds a fortiori for diagonal matrices.

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 which is equal to its own conjugate transpose - that is, the element in the ith row and jth column is equal to the complex conjugate of the element in the jth row and ith column, for... In linear algebra, a positive-definite matrix is a Hermitian matrix which in many ways is analogous to a 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 an 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... In mathematics, the absolute value (or modulus1) 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, the main diagonal of a square matrix is the diagonal which runs from the top left corner to the bottom right corner. ... 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, a determinant is a function depending on n that associates a scalar det(A) to every n×n square matrix A. The fundamental geometric meaning of a 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.

Conjugate eigenvector

A conjugate eigenvector or coneigenvector is a vector sent after transformation to a scalar multiple of its conjugate, where the scalar is called the conjugate eigenvalue or coneigenvalue of the linear transformation. The coneigenvectors and coneigenvalues represent essentially the same information and meaning as the regular eigenvectors and eigenvalues, but arise when a alternative coordinate system is used. The corresponding equation is

For example, in coherent electromagnetic scattering theory, the linear transformation A represents the action performed by the scattering object, and the eigenvectors represent polarization states of the electromagnetic wave. In optics, the coordinate system is defined from the wave's viewpoint, known as the Forward Scattering Alignment (FSA), and gives rise to a regular eigenvalue equation, whereas in radar, the coordinate system is defined from the radar's viewpoint, known as the Back Scattering Alignment (BSA), and gives rise to a coneigenvalue equation. Optics (appearance or look in ancient Greek) is a branch of physics that describes the behavior and properties of light and the interaction of light with matter. ... The Forward Scattering Alignment (FSA) is a coordinate system used in coherent electromagnetic scattering. ... This long range radar antenna (approximately 40m (130ft) in diameter) rotates on a track to observe activities near the horizon. ... The Back Scattering Alignment (BSA) is a coordinate system used in coherent electromagnetic scattering. ...

Generalized eigenvalue problem

A generalized eigenvalue problem (2nd sense) is of the form

where A and B are matrices. The generalized eigenvalues (2nd sense) λ can be obtained by solving the equation

If B is invertible, then the original problem can be 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 the eigenvalues are real. This would have not been easy to see from the second equivalent formulation, because the matrix B ^{− 1}A is not necessarily symmetric if A and B are.

An example is provided by the molecular orbital application below. In this transformation of the Mona Lisa, the blue vector has been rotated, but the red one has not. ...

Eigenvalues of a matrix with entries from a ring

In the case of a square matrix A with entries in a ring, λ is called a right eigenvalue if there exists a 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. ... In linear algebra, a column vector is an m x 1 matrix, a matrix consisting of a single column. ... In linear algebra, a row vector is a 1 × n matrix, i. ...

If the ring is commutative, the left eigenvalues are equal to the right eigenvalues and are just called eigenvalues. If not, for instance if the ring is a set of quaternions, they may be different. In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... In mathematics, the quaternions are a non-commutative extension of the complex numbers. ...

Linear involutions and projections

Involutions, that is transformations such that , and projections, that is transformations such that , if they are linear, have the following simple properties In mathematics, an involution is a function that is its own inverse, so that f(f(x)) = x for all x in the domain of f. ... The word projection can mean more than one thing. ...

• The eigenvalues of an involution can only be 1 or −1.
• The eigenvalues of a projection operator can only be 0 or 1.

Such kind of transformations are ubiquitous in mathematics and applied disciplines. The simplest examples are the identity () which is an involution as well as a projection, the inversion () which is an involution and the zero operator ().

Infinite-dimensional spaces: spectrum of an operator

Absorption spectrum (cross section) of atomic Chlorine. The sharp lines obtained in theory correspond to the discrete spectrum (Rydberg series) of the Hamiltonian the broad structure on the right is associated to the continuous spectrum (ionization). The associated experimental results have been obtained by measuring the intensity of X-rays absorbed by a gas of atoms as a function of the incident photon energy in eV.^[1]

Main article: spectrum of an operator

If the vector space is infinite dimensional, it may be advantageous to define the concept of spectral values. The spectral values are the set of scalars λ for which the Green's operator associated to the transformation is not defined, that is such that is not invertible (i.e., the inverse transformation to does not exists). Image File history File links Discrete-continuum. ... Image File history File links Discrete-continuum. ... Absorption spectroscopy is an analytical tool used by chemists. ... Spectrum of fluorescent lights showing prominent mercury peaks. ... Cross section may refer to the following In geometry, Cross section is the intersection of a 3-dimensional body with a plane. ... General Name, Symbol, Number chlorine, Cl, 17 Chemical series halogens Group, Period, Block 17, 3, p Appearance yellowish green Atomic mass 35. ... In mathematics and physics, discrete spectrum is a finite set or a countable set of eigenvalues of an operator. ... A Rydberg atom is an excited atom with one or more electrons that have a very high principal quantum number. ... The Hamiltonian, denoted H, has two distinct but closely related meanings. ... In mathematics and physics, continuous spectrum is, roughly speaking, a non-countable set of eigenvalues of an operator. ... ... From Latin ex- + -periri (akin to periculum attempt). ... In the NATO phonetic alphabet, X-ray represents the letter X. An X-ray picture (radiograph) taken by Röntgen An X-ray is a form of electromagnetic radiation with a wavelength approximately in the range of 5 pm to 10 nanometers (corresponding to frequencies in the range 30 PHz... For the Science Fiction weapon, as seen in Star Trek, see Photon torpedo. ... eV may stand for: electronvolt eingetragener Verein, i. ... 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 λ is an eigenvalue of , λ is also a spectral value of it. However, the reverse relation is not true: any spectral value is not an eigenvalue. There are operators on Hilbert or Banach spaces which have no eigenvectors at all. This can be seen on the following example. The bilateral shift on the Hilbert space (the space of all sequences of scalars such that converge) has no eigenvalue but has spectral values. 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. ... In mathematics, and in particular functional analysis, the shift operators are examples of linear operators, important for their simplicity and natural occurrence. ...

In functional analysis, the spectrum of an operator is defined as the set of all its spectral values. This is a key concept in scattering theory. Functional analysis is that branch of mathematics and specifically of analysis which is concerned with the study of spaces of functions. ... Scattering theory is a branch of physics and especially of quantum mechanics whose aim is the study of scattering events. ...

In infinite-dimensional spaces, the spectrum of an operator can be discrete or continuous. The former case occurs if the spectrum is a countable set of scalars; the latter if not. The exponential growth or decay provides an example of a continuous spectrum and the vibrating string an example of a discrete one (see above). The hydrogen atom is an example where both type of spectra appear (see below). The bound states of the hydrogen atom correspond to the discrete part of the spectrum while the ionization processes are described by the continuous part. In mathematics and physics, discrete spectrum is a finite set or a countable set of eigenvalues of an operator. ... In mathematics and physics, continuous spectrum is, roughly speaking, a non-countable set of eigenvalues of an operator. ... In mathematics the term countable set is used to describe the size of a set, e. ... In this transformation of the Mona Lisa, the blue vector has been rotated, but the red one has not. ... A hydrogen atom is an atom of the element hydrogen. ... In this transformation of the Mona Lisa, the blue vector has been rotated, but the red one has not. ... In physics, a bound state is a composite of two or more building blocks (particles) that behaves as a single object. ... ...

Applications

Schrödinger equation

The wavefunctions associated to the bound states of an electron in a hydrogen atom can be seen as the eigenvectors of the hydrogen atom Hamiltonian as well as of the angular momentum operator. They are associated to eigenvalues interpreted as their energies (increasing downward: n=1,2,3,...) and angular momentum (increasing across: s, p, d,...). Here are plotted the square of the absolute value of the wavefunctions. Brighter areas correspond to higher probability density for a position measurement. The center of each figure is the atomic nuclei, a proton

An example of eigenvalue equation where the transformation is represented in terms of a differential operator is the time-independent Schrödinger equation in quantum mechanics First few hydrogen atom orbitals; cross section showing color-coded probability density for different n=1,2,3 and l=s,p,d; note: m=0 The picture shows the first few hydrogen atom orbitals (energy eigenfunctions). ... First few hydrogen atom orbitals; cross section showing color-coded probability density for different n=1,2,3 and l=s,p,d; note: m=0 The picture shows the first few hydrogen atom orbitals (energy eigenfunctions). ... In quantum mechanics, the wavefunction associated with a particle such as an electron, is a complex-valued function ψ defined over a portion of space and normalized in such a way that In Max Borns probabilistic interpretation of the wavefunction, the amplitude squared of the wavefunction |ψ(x)|2 is the... In physics, a bound state is a composite of two or more building blocks (particles) that behaves as a single object. ... Properties The electron is a fundamental subatomic particle which carries a negative electric charge. ... A hydrogen atom is an atom of the element hydrogen. ... A hydrogen atom is an atom of the element hydrogen. ... In physics the angular momentum of an object with respect to a reference point is a measure for the extent to which, and the direction in which, the object rotates about the reference point. ... In physics the angular momentum of an object with respect to a reference point is a measure for the extent to which, and the direction in which, the object rotates about the reference point. ... In quantum mechanics, a probability amplitude is a complex number-valued function which describes an uncertain or unknown quantity. ... The framework of quantum mechanics requires a careful definition of measurement, and a thorough discussion of its practical and philosophical implications. ... Plural: nuclei In chemistry and physics, the nucleus (atomic nucleus) is the collection of protons and neutrons in the center of an atom that carries the bulk of the atoms mass and positive charge. ... Properties In physics, the proton (Greek proton = first) is a subatomic particle with an electric charge of one positive fundamental unit (1. ... In physics, the Schrödinger equation, proposed by the Austrian physicist Erwin Schrödinger in 1925, describes the time-dependence of quantum mechanical systems. ... Fig. ...

HΨ_E = EΨ_E

where H, the Hamiltonian, is a second-order differential operator and Ψ_E, the wavefunction, is one of its eigenfunctions corresponding to the eigenvalue E, interpreted as its energy. The Hamiltonian, denoted H, has two distinct but closely related meanings. ... In mathematics, a differential operator is a linear operator defined as a function of the differentiation operator. ... In quantum mechanics, the wavefunction associated with a particle such as an electron, is a complex-valued function ψ defined over a portion of space and normalized in such a way that In Max Borns probabilistic interpretation of the wavefunction, the amplitude squared of the wavefunction |ψ(x)|2 is the... In this transformation of the Mona Lisa, the blue vector has been rotated, but the red one has not. ...

However, in the case we only look for the bound state solutions of the Schrödinger equation, as is usually the case in quantum chemistry, we look for Ψ_E within the space of square integrable functions. Since this space is a Hilbert space, with a well-defined scalar product, we can introduce a basis set in which Ψ_E and H can be represented as a one-dimensional array and a matrix respectively. This allows us to represent the Schrödinger equation in a matrix form. In physics, a bound state is a composite of two or more building blocks (particles) that behaves as a single object. ... Quantum chemistry is a branch of theoretical chemistry, which applies quantum mechanics to problems in chemistry. ... In mathematical analysis, a real- or complex-valued function of a real variable is square-integrable on an interval if the integral over that interval of the square of its absolute value is finite. ... 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, the dot product (also known as the scalar product and the inner product) is a function (·) : V × V → F, where V is a vector space and F its underlying field. ... 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...

The Dirac notation often used in this context stresses the difference between the vector or state and its representation, the function Ψ_E. In this context one writes the Schrödinger equation Bra-ket notation is the standard notation for describing quantum states in the theory of quantum mechanics. ...

and call an eigenstate of H (sometimes written in introductory textbooks) which is seen as a transformation (see Observable) instead of particular representation of it in terms of differential operators. In the equation above is understood as the vector obtained by application of the transformation H to . 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. ...

Molecular orbitals

In quantum mechanics, and in particular in atomic and molecular physics, within the Hartree-Fock theory, the atomic and molecular orbitals can be defined by the eigenvectors of the Fock operator. The corresponding eigenvalues are interpreted as ionization potentials via Koopmans' theorem. In this case, the term eigenvector is used in a somewhat more general meaning, since the Fock operator is explicitly dependent of the orbitals and their eigenvalues. If one wants to underline this aspect one speaks of implicit eigenvalue equation. Such equations are usually solved by an iteration procedure, called in this case self-consistent field method. In quantum chemistry, one often represents the Hartree-Fock equation in a non-orthogonal basis set. This particular representation is a generalized eigenvalue problem called Roothaan equations. Fig. ... Atomic physics (or atom physics) is physics of the electron hull of atoms. ... Molecular physics is the study of the physical properties of molecules and of the chemical bonds between atoms that bind them into molecules. ... In computational physics and computational chemistry, the Hartree-Fock (HF) or self-consistent field (SCF) calculation scheme is a self-consistent iterative variational procedure to calculate the Slater determinant (or the molecular orbitals which it is made of) for which the expectation value of the electronic molecular Hamiltonian is minimum. ... Electron atomic and molecular orbitals In quantum mechanics, the states of an atom, i. ... Electron atomic and molecular orbitals In quantum chemistry, the molecular electronic states, i. ... In quantum mechanics, the Fock matrix is a matrix approximating the single-electron energy operator of a given quantum system in a given set of basis vectors. ... The ionization potential, or ionization energy, of an atom of a molecule is the energy required to strip it of an electron. ... Koopmans theorem is an approximation in molecular orbital theory, such as density functional theory, or Hartree-Fock theory, in which the first ionization energy of a molecule is equal to the energy of the highest occupied molecular orbital (the HOMO), and the electron affinity is the negative of the energy... Iteration is the repetition of a process, typically within a computer program. ... In computational physics and computational chemistry, the Hartree-Fock (HF) or self-consistent field (SCF) calculation scheme is a self-consistent iterative variational procedure to calculate the Slater determinant (or the molecular orbitals which it is made of) for which the expectation value of the electronic molecular Hamiltonian is minimum. ... Quantum chemistry is a branch of theoretical chemistry, which applies quantum mechanics to problems in chemistry. ... 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. ... In modern computational chemistry, quantum chemical calculations are typically performed within a finite set of basis functions. ... In this transformation of the Mona Lisa, the blue vector has been rotated, but the red one has not. ... The Roothaan equations are a representation of the Hartree-Fock equation in a non orthonormalized basis set which can be of Gaussian type. ...

Factor analysis

In factor analysis, the eigenvectors of a covariance matrix correspond to factors, and eigenvalues to factor loadings. Factor analysis is a statistical technique used in the social sciences and in marketing, product management, operations research, and other applied sciences that deal with large quantities of data. The objective is to explain most of the variability among a number of observable random variables in terms of a smaller number of unobservable random variables called factors. The observable random variables are modeled as linear combinations of the factors, plus "error" terms. Factor analysis is a statistical technique that originated in mathematical psychology. ... In statistics and probability theory, the covariance matrix is a matrix of covariances between elements of a vector. ... Factor analysis is a statistical technique that originated in mathematical psychology. ... Factor analysis is a statistical technique that originated in mathematical psychology. ... Statistics is a type of data analysis which practice includes the planning, summarizing, and interpreting of observations of a system possibly followed by predicting or forecasting of future events based on a mathematical model of the system being observed. ... Terms like SOSE (Studies of Society & the Environment) not only refer to social sciences but also studies of the environment. ... Marketing is the process of planning and executing the pricing, promotion, and distribution of goods, ideas, and services to create exchanges that satisfy individual and organizational goals. ... Product Management is one of the four areas of marketing. ... Operations research, operational research, or simply OR, is the use of mathematical models, statistics and algorithms to aid in decision-making. ... A random variable can be thought of as the numeric result of operating a non-deterministic mechanism or performing a non-deterministic experiment to generate a random result. ... In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ... In statistics, the concepts of error and residual are easily confused with each other. ...

Eigenfaces as examples of eigenvectors

Image File history File links Eigenfaces of faces from the ORL face database. ... Image File history File links Eigenfaces of faces from the ORL face database. ... Eigenfaces are a set of eigenvectors derived from the covariance matrix of the probability distribution of the high-dimensional vector space of possible faces of human beings. ...

Eigenfaces

In image processing, processed images of faces can be seen as vectors whose components are the brightnesses of each pixels. The dimension of this vector space is the number of pixels. The eigenvectors of the covariance matrix associated to a large set of normalized pictures of faces are called eigenfaces. They are very useful for expressing any face image as a linear combination of some of them. Eigenfaces provide a means of applying data compression to faces for identification purposes. This article needs to be cleaned up to conform to a higher standard of quality. ... This article may be too technical for most readers to understand. ... Brightness is an attribute of visual perception in which a source appears to emit a given amount of light. ... A pixel (a portmanteau of picture element) is one of the many tiny dots that make up the representation of a picture in a computers memory. ... In statistics and probability theory, the covariance matrix is a matrix of covariances between elements of a vector. ... Eigenfaces are a set of eigenvectors derived from the covariance matrix of the probability distribution of the high-dimensional vector space of possible faces of human beings. ... In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ... In computer science, data compression or source coding is the process of encoding information using fewer bits (or other information-bearing units) than a more obvious representation would use, through use of specific encoding schemes. ...

Tensor of inertia

In mechanics, the eigenvectors of the inertia tensor define the principal axes of a rigid body. The tensor of inertia is a key quantity required in order to determine the rotation of a rigid body around its center of mass. Mechanics refers to: a craft relating to machinery (from the Latin mechanicus, from the Greek mechanikos, meaning one skilled in machines), or a range of disciplines in science and engineering. ... Moment of inertia quantifies the resistance of a physical object to angular acceleration. ... In mathematics, particularly linear algebra and functional analysis, the spectral theorem is a collection of results about linear operators or about matrices. ... In physics, a rigid body is an idealisation of a solid body of finite dimension in which deformation is neglected. ... The center of mass or center of inertia of an object is a point at which the objects mass can be assumed, for many purposes, to be concentrated. ...

Stress tensor

In solid mechanics, the stress tensor is symmetric and so can be decomposed into a diagonal tensor with the eigenvalues on the diagonal and eigenvectors as a basis. Because it is diagonal, in this orientation, the stress tensor has no shear components; the components it does have are the principal components. Solid mechanics (also known as the theory of elasticity) is a branch of physics, which governs the response of solid material to applied stress (e. ... This article is in need of attention from an expert on the subject. ... In mathematics, diagonal has a geometric meaning, and a derived meaning as used in square tables and matrix terminology. ... In physics and mechanics, shear refers to a deformation that causes parallel surfaces to slide past one another (as opposed to compression and tension, which cause parallel surfaces to move towards or away from one another). ...

Eigenvalues of a graph

In spectral graph theory, 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 / 2}AT ^{− 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, spectral graph theory is the study of properties of a graph in relationship to the eigenvalues and eigenvectors of its adjacency matrix. ... 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 . ...

External links

• Videos of MIT Linear Algebra Course, fall 1999 - See Lecture #21: Eigenvalues and Eigenvectors
• MathWorld: Eigenvector
• Earliest Known Uses of Some of the Words of Mathematics: E - see eigenvector and related terms
• ARPACK is a collection of FORTRAN subroutines for solving large scale eigenvalue problems
• Eigenvalue (of a matrix) on PlanetMath.

Wikibooks Algebra has more about this subject:

Eigenvalues and eigenvectors

PlanetMath is a free, collaborative, online mathematics encyclopedia. ... Wikipedia does not have an article with this exact name. ...

References

• Roger A. Horn and Charles R. Johnson, Matrix Analysis, Cambridge University Press (1985). ISBN 0-521-30586-1 (hardback), ISBN 0-521-38632-2 (paperback).
• John B. Fraleigh and Raymond A. Beauregard, Linear Algebra (3^rd edition), Addison-Wesley Publishing Company (1995). ISBN 0-201-83999-7 (international edition).
• Claude Cohen-Tannoudji, Quantum Mechanics, Wiley (1977). ISBN 0-471-16432-1. (Chapter II. The mathematical tools of quantum mechanics.)

Claude Cohen-Tannoudji (born April 1, French physicist working at the École Normale Supérieure in Paris, France, where he has also studied physics. ...

Notes

1. ^  Vectors pointing from one point to the same point (zero vectors) are usually not considered as eigenvectors.
2. ^  T. W Gorczyca, Auger Decay of the Photoexcited Inner Shell Rydberg Series in Neon, Chlorine, and Argon, Abstracts of the 18th International Conference on X-ray and Inner-Shell Processes, Chicago, August 23-27 (1999).