Eigenvalues and eigenvectors

For more specific information regarding the eigenvalues and eigenvectors of matrices, see Eigendecomposition (matrix).

Fig. 1. In this shear mapping of the Mona Lisa, the picture was deformed in such a way that its central vertical axis (red vector) was not modified, but the diagonal vector (blue) has changed direction. Hence the red vector is an eigenvector of the transformation and the blue vector is not. Since the red vector was neither stretched nor compressed, its eigenvalue is 1. All vectors with the same vertical direction - i.e., parallel to this vector - are also eigenvectors, with the same eigenvalue. Together with the zero-vector, they form the eigenspace for this eigenvalue.

In mathematics, a vector may be thought of as an arrow. It has a length, called its magnitude, and it points in some particular direction. A linear transformation may be considered to operate on a vector to change it, usually changing both its magnitude and its direction. An of a given linear transformation is a vector which is multiplied by a constant called the during that transformation. The direction of the eigenvector is either unchanged by that transformation (for positive eigenvalues) or reversed (for negative eigenvalues).

For example, an eigenvalue of +2 means that the eigenvector is doubled in length and points in the same direction. An eigenvalue of +1 means that the eigenvector stays the same, while an eigenvalue of −1 means that the eigenvector is reversed in direction. An eigenspace of a given transformation is the set of all eigenvectors of that transformation that have the same eigenvalue, together with the zero vector (which has no direction). An eigenspace is an example of a subspace of a vector space.

In linear algebra, every linear transformation between finite-dimensional vector spaces can be given by a matrix, which is a rectangular array of numbers arranged in rows and columns. Standard methods for finding eigenvalues, eigenvectors, and eigenspaces of a given matrix are discussed below.

These concepts play a major role in several branches of both pure and applied mathematics — appearing prominently in linear algebra, functional analysis, and to a lesser extent in nonlinear mathematics.

Many kinds of mathematical objects can be treated as vectors: functions, harmonic modes, quantum states, and frequencies, for example. In these cases, the concept of direction loses its ordinary meaning, and is given an abstract definition. Even so, if this abstract direction is unchanged by a given linear transformation, the prefix "eigen" is used, as in eigenfunction, eigenmode, eigenstate, and eigenfrequency.

History

Eigenvalues are often introduced in the context of linear algebra or matrix theory. Historically, however, they arose in the study of quadratic forms and differential equations.

Euler had also studied the rotational motion of a rigid body and discovered the importance of the principal axes. As Lagrange realized, the principal axes are the eigenvectors of the inertia matrix. In the early 19th century, Cauchy saw how their work could be used to classify the quadric surfaces, and generalized it to arbitrary dimensions. Cauchy also coined the term racine caractéristique (characteristic root) for what is now called eigenvalue; his term survives in characteristic equation.

Fourier used the work of Laplace and Lagrange to solve the heat equation by separation of variables in his famous 1822 book Théorie analytique de la chaleur. Sturm developed Fourier's ideas further and he brought them to the attention of Cauchy, who combined them with his own ideas and arrived at the fact that symmetric matrices have real eigenvalues. This was extended by Hermite in 1855 to what are now called Hermitian matrices. Around the same time, Brioschi proved that the eigenvalues of orthogonal matrices lie on the unit circle, and Clebsch found the corresponding result for skew-symmetric matrices. Finally, Weierstrass clarified an important aspect in the stability theory started by Laplace by realizing that defective matrices can cause instability.

In the meantime, Liouville had studied similar eigenvalue problems as Sturm; the discipline that grew out of their work is now called Sturm-Liouville theory. Schwarz studied the first eigenvalue of Laplace's equation on general domains towards the end of the 19th century, while Poincaré studied Poisson's equation a few years later.

At the start of the 20th century, Hilbert studied the eigenvalues of integral operators by considering them to be infinite matrices. He was the first to use the German word eigen to denote eigenvalues and eigenvectors in 1904, though he may have been following a related usage by Helmholtz. "Eigen" can be translated as "own", "peculiar to", "characteristic" or "individual"—emphasizing how important eigenvalues are to defining the unique nature of a specific transformation. For some time, the standard term in English was "proper value", but the more distinctive term "eigenvalue" is standard today.

The first numerical algorithm for computing eigenvalues and eigenvectors appeared in 1929, when Von Mises published the power method. One of the most popular methods today, the QR algorithm, was proposed independently by Francis and Kublanovskaya in 1961.

Definitions

See also: Eigenplane

Linear transformations of a vector space, such as rotation, reflection, stretching, compression, shear or any combination of these, may be visualized by the effect they produce on vectors, in other words, those are vector functions. Said more formally, in vector space L a vector function A with vector argument x is defined if to each vector x of L corresponds some vector y = A(x) of L. For the sake of brevity, the parentheses around the vector on which the transformation is acting, are often omitted. Vector function A is defined as linear if it has the following two properties:

additivity $\ A(\mathbf {x} +\mathbf {y} )=A(\mathbf {x} )+A(\mathbf {y} )$

homogeneity $\ A(\alpha \mathbf {x} )=\alpha A(\mathbf {x} )$

where x and y are any two vectors of the vector space L and α is any real number. Such linear vector function is called linear transformation of space L, linear operator, or linear endomorphism in this space.

Definition: Given the linear transformation Ax = y, the non-zero vector x is defined as an eigenvector of this transformation if it satisfies the eigenvalue equation $A\mathbf {x} =\lambda \mathbf {x}$ where λ is a scalar defined as an eigenvalue of vector x.

The key equation in this definition is the eigenvalue equation, Ax = λx. Most vectors x will not satisfy such an equation. A typical x changes direction when multiplied by A, so that Ax is not a multiple of x. This means that only certain special numbers λ are eigenvalues, and only certain special vectors x are eigenvectors. Of course, if A were a multiple of the identity matrix, then no vector would change direction, and all vectors would be eigenvectors. But in the usual case, eigenvectors are few and far between. They are the "normal modes" of the system, and they act independently.

The requirement for the eigenvector not to be zero is because if x = 0, the eigenvalue equation will become 0 = 0. Since this equation is always true, it is not an interesting case, it is a trivial case. Therefore, the eigenvector is defined to be a non-zero vector that satisfies the eigenvalue equation, Ax = λx. However, an eigenvalue can be zero without causing a problem. An eigenvalue can be (and usually is) also a complex number. Somewhat more informally, x is defined as an eigenvector corresponding to the eigenvalue λ if they satisfy Ax = λx. Since each eigenvector is associated with an eigenvalue, an x and λ that correspond to one another are often referred to as an eigenpair.

Geometrically, the eigenvalue equation means that under the transformation A eigenvectors experience only changes in magnitude and sign — the orientation of Ax is the same as that of x. This type of linear transformation is defined as homothety (dilatation, similarity transformation). The eigenvalue λ is simply the amount of "stretch" or "shrink" to which the eigenvector is subjected when transformed by A. If λ = 1, the eigenvector remains unchanged (unaffected by the transformation). A transformation I under which a vector x remains unchanged, Ix = x, is defined as identity transformation. If λ = –1, the eigenvector flips to the opposite direction (rotates to 180°); this is defined as reflection.

If x is an eigenvector of the linear transformation A, then any collinear (on the same straight line) vector x' = αx (α is a non-zero real number) is also an eigenvector of the linear transformation A with the same eigenvalue λ as x. Indeed, because of the homogeneity of the transformation A it follows that Ax' = A(αx) = α(Ax) = α(λx) = λ(αx) = λx'. Similarly, using the additivity property of the linear transformation, it can be shown that any linear combination of eigenvectors with an eigenvalue λ has the same eigenvalue λ. Therefore, there are infinitely many eigenvectors with one and the same eigenvalue; together with the zero-vector those eigenvectors form a subspace of the respective vector space, defined as eigenspace. The eigenvectors, corresponding to different eigenvalues, are linearly independent meaning, in particular, that in an n-dimensional space the linear transformation A cannot have more than n eigenvectors with different eigenvalues. The vectors of the eigenspace generate a linear subspace of A which is invariant (unchanged) under this transformation.

If a basis is defined in vector space L_n, all vectors can be expressed in terms of components. Polar vectors can be represented as one-column matrices with n rows where n is the space dimensionality. Linear transformations can be represented with square matrices; to each linear transformation A of L_n corresponds a square matrix of rank n. Conversely, to each square matrix of rank n corresponds a linear transformation of L_n at a given basis. Because of the additivity and homogeneity of the linear trasformation and the eigenvalue equation (which is also a linear transformation — homothety), those vector functions can be expressed in matrix form. Thus, in a the two-dimensional vector space L₂ fitted with standard basis, the eigenvector equation for a linear transformation A can be written in the following matrix representation:

${\begin{bmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{bmatrix}}{\begin{bmatrix}x\\y\end{bmatrix}}=\lambda {\begin{bmatrix}x\\y\end{bmatrix}},$

where the juxtaposition of matrices means 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 these equations both the eigenvalue λ and the components of x are unknown variables.

The applicability the eigenvalue equation to the general matrix theory extends the use of eigenvectors and eigenvalues to all matrices, and thus greatly extends the scope of use of these mathematical constructs not only to transformations in linear vector spaces but to all fields of science that use matrices: linear equations systems, optimisation, vector and tensor calculus, all fields of physics that use matrix quantities, particularly quantum physics, relativity, and electrodynamics, as well as many engineering applications.

An eigenvector x is called a principal eigenvector (dominant eigenvector) if it corresponds to an eigenvalue λ that is maximal among all eigenvalues.

Characteristic equation

Main article: Characteristic equation

Main article: Characteristic polynomial

The determination of the eigenvalues and eigenvectors is important in virtually all areas of physics and many engineering problems, such as stress calculations, stability analysis, oscillations of vibrating systems, etc. It is equivalent to matrix diagonalization, and serves as a first stage of system orthogonalization, finding of invariants, system optimization (minimization or maximization), analysis of linear systems, and many other common applications.

The usual method of finding all eigenvectors and eigenvalues of a system is first to get rid of the unknown components of the eigenvectors, then find the eigenvalues, plug those back one by one in the eigenvalue equation in matrix form and solve that as a system of linear equations to find the components of the eigenvectors. From the identity transformation Ix = x, where I is the identity matrix, x in the eigenvalue equation can be replaced by Ix to give:

$A\mathbf {x} =\lambda I\mathbf {x}$

The identity matrix is needed to keep matrices, vectors, and scalars straight; the equation (A − λ) x = 0 is shorter, but mixed up since it does not differentiate between matrix, scalar, and vector. The expression in the right hand side is transferred to left hand side with a negative sign, leaving 0 on the right hand side:

$A\mathbf {x} -\lambda I\mathbf {x} =0$

The eigenvector x is pulled out behind parentheses:

$(A-\lambda I)\mathbf {x} =0$

This can be viewed as a linear system of equations in which the coefficient matrix is the expression in the parentheses, the matrix of the unknowns is x, and the right hand side matrix is zero. According to Cramer's rule, this system of equations has non-trivial solutions (not all zeros, or not any number) if and only if its determinant vanishes, so the solutions of the equation are given by:

$\det(A-\lambda I)=0\,$

This equation is defined as the characteristic equation (less often, secular equation) of A, and the left-hand side is defined as the characteristic polynomial. The eigenvector x or its components are not present in the characteristic equation, so at this stage they are dispensed with, and the only unknowns that remain to be calculated are the eigenvalues (the components of matrix A are given, i. e, known beforehand). For a vector space L₂, the transformation A is a 2 × 2 square matrix, and the characteristic equation can be written in the following form:

${\begin{vmatrix}a_{11}-\lambda &a_{12}\\a_{21}&a_{22}-\lambda \end{vmatrix}}=0$

Expansion of the determinant in the left hand side results in a characteristic polynomial which is a monic (its leading coefficient is 1) polynomial of the second degree, and the characteristic equation is the quadratic equation

$\lambda ^{2}-\lambda (a_{11}+a_{22})+(a_{11}a_{22}-a_{12}a_{21})=0,\,$

which has the following solutions (roots):

$\lambda _{1,2}={\frac {1}{2}}\left[(a_{11}+a_{22})\pm {\sqrt {4a_{12}a_{21}+(a_{11}-a_{22})^{2}}}\right].$

For real matrices, the coefficients of the characteristic polynomial are all real. The number and type of roots depends on the value of the discriminant, Δ. For cases Δ = 0, Δ > 0, or Δ < 0, respectively, the roots are one real, two real, or two complex. If the roots are complex, they are also complex conjugates of each other. When the number of roots is less than the degree of the characteristic polynomial (the latter is also the rank of the matrix, and the number of dimensions of the vector space) the equation has a multiple root. In the case of a quadratic equation with one root, this root is a double root, or a root with multiplicity 2. A root with a multiplicity of 1 is a simple root. A quadratic equation with two real or complex roots has only simple roots. In general, the algebraic multiplicity of an eigenvalue is defined as the multiplicity of the corresponding root of the characteristic polynomial. The spectrum of a transformation on a finite dimensional vector space is defined as the set of all its eigenvalues. In the infinite-dimensional case, the concept of spectrum is more subtle and depends on the topology of the vector space.

The general formula for the characteristic polynomial of an n-square matrix is

$p(\lambda )=\sum _{k=0}^{n}(-1)^{k}S_{k}\lambda ^{n-k},$

where S₀ = 1, S₁ = tr(A), the trace of the transformation matrix A, and S_k with k > 1 are the sums of the principal minors of order k. The fact that eigenvalues are roots of an n-order equation shows that a linear transformation of an n-dimensional linear space has at most n different eigenvalues. According to the fundamental theorem of algebra, in a complex linear space, the characteristic polynomial has at least one zero. Consequently, every linear transformation of a complex linear space has at least one eigenvalue. For real linear spaces, if the dimension is an odd number, the linear transformation has at least one eigenvalue; if the dimension is an even number, the number of eigenvalues depends on the determinant of the transformation matrix: if the determinant is negative, there exists at least one positive and one negative eigenvalue, if the determinant is positive nothing can be said about existence of eigenvalues. The complexity of the problem for finding roots/eigenvalues of the characteristic polynomial increases rapidly with increasing the degree of the polynomial (the dimension of the vector space), n. Thus, for n = 3, eigenvalues are roots of the cubic equation, for n = 4 — roots of the quartic equation. For n > 4 there are no exact solutions and one has to resort to root-finding algorithms, such as Newton's method (Horner's method) to find numerical approximations of eigenvalues. For large symmetric sparse matrices, Lanczos algorithm is used to compute eigenvalues and eigenvectors.

• The geometric multiplicity of an eigenvalue is the dimension of the associated eigenspace.

For instance, an eigenvector of a rotation in three dimensions is a vector located within the axis about which the rotation is performed. The corresponding eigenvalue is 1 and the corresponding eigenspace contains all the vectors along 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.

Examples

Mona Lisa

For the example shown on the right, the matrix that would produce a shear transformation similar to this would be

$A={\begin{bmatrix}1&0\\-{\frac {1}{2}}&1\end{bmatrix}}.$

The set of eigenvectors $\mathbf {x}$ for $A$ is defined as those vectors which, when multiplied by $A$, result in a simple scaling $\lambda$ of $\mathbf {x}$. Thus,

$A\mathbf {x} =\lambda \mathbf {x} .$

If we restrict ourselves to real eigenvalues, the only effect of the matrix on the eigenvectors will be to change their length, and possibly reverse their direction. So multiplying the right hand side by the Identity matrix I, we have

$A\mathbf {x} =(\lambda I)\mathbf {x} ,$

and therefore

$(A-\lambda I)\mathbf {x} =0.$

In order for this equation to have non-trivial solutions, we require the determinant $\det(A-\lambda I)$, which is called the characteristic polynomial of the matrix A, to be zero. In our example we can calculate the determinant as

$\det \!\left({\begin{bmatrix}1&0\\-{\frac {1}{2}}&1\end{bmatrix}}-\lambda {\begin{bmatrix}1&0\\0&1\end{bmatrix}}\right)=(1-\lambda )^{2},$

and now we have obtained the characteristic polynomial $(1-\lambda )^{2}$ of the matrix A. There is in this case only one distinct solution of the equation $(1-\lambda )^{2}=0$, $\lambda =1$. This is the eigenvalue of the matrix A. As in the study of roots of polynomials, it is convenient to say that this eigenvalue has multiplicity 2.

Having found an eigenvalue $\lambda =1$, we can solve for the space of eigenvectors by finding the nullspace of $A-(1)I$. In other words by solving for vectors $\mathbf {x}$ which are solutions of

${\begin{bmatrix}1-\lambda &0\\-{\frac {1}{2}}&1-\lambda \end{bmatrix}}{\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}=0$

Substituting our obtained eigenvalue $\lambda =1$,

${\begin{bmatrix}0&0\\-{\frac {1}{2}}&0\end{bmatrix}}{\begin{bmatrix}x_{1}\\x_{2}\end{bmatrix}}=0$

Solving this new matrix equation, we find that vectors in the nullspace have the form

$\mathbf {x} ={\begin{bmatrix}0\\c\end{bmatrix}}$

where c is an arbitrary constant. All vectors of this form, i.e. pointing straight up or down, are eigenvectors of the matrix A. The effect of applying the matrix A to these vectors is equivalent to multiplying them by their corresponding eigenvalue, in this case 1.

In general, 2-by-2 matrices will have two distinct eigenvalues, and thus two distinct eigenvectors. Whereas most vectors will have both their lengths and directions changed by the matrix, eigenvectors will only have their lengths changed, and will not change their direction, except perhaps to flip through the origin in the case when the eigenvalue is a negative number. Also, it is usually the case that the eigenvalue will be something other than 1, and so eigenvectors will be stretched, squashed and/or flipped through the origin by the matrix.

Other examples

As the Earth rotates, every arrow pointing outward from the center of the Earth also rotates, except those arrows which are parallel to 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 Geographic South Pole would be an eigenvector of this transformation, but an arrow from the center of the Earth to anywhere on the equator would not be an eigenvector. Since the arrow pointing at the pole is not stretched by the rotation of the Earth, its eigenvalue is 1.

Another example is provided by a rubber sheet expanding omnidirectionally about a fixed point in such a way that the distances from any point of the sheet to the fixed point are doubled. This expansion is a transformation with 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.

Fig. 2. A standing wave in a rope fixed at its boundaries is an example of an eigenvector, or more precisely, an eigenfunction of the transformation giving the acceleration. As time passes, the standing wave is scaled by a sinusoidal oscillation whose frequency is determined by the eigenvalue, but its overall shape is not modified.

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 (Fig. 2). 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 there are atoms in the rope.

Assume the rope is a continuous medium. If one considers the equation for the acceleration at every point of the rope, its eigenvectors, or eigenfunctions, are the standing waves. The standing waves correspond to particular oscillations of the rope such that the acceleration of the rope is simply its shape scaled by a factor—this factor, the eigenvalue, turns out to be $-\omega ^{2}$ where $\omega$ is the angular frequency of the oscillation. Each component of the vector associated with the rope is multiplied by a time-dependent factor $\sin(\omega t)$. If damping is considered, the amplitude of this oscillation decreases until the rope stops oscillating, corresponding to a complex ω. One can then associate a lifetime with the imaginary part of ω, and relate the concept of an eigenvector to the concept of resonance. Without damping, the fact that the acceleration operator (assuming a uniform density) is Hermitian leads to several important properties, such as that the standing wave patterns are orthogonal functions.

Eigenfunctions

However, it is sometimes unnatural or even impossible to write down the eigenvalue equation in a matrix form. This occurs for instance when the vector space is infinite dimensional, for example, in the case of the rope above. Depending on the nature of the transformation T and the space to which it applies, it can be advantageous to represent the eigenvalue equation as a set of differential equations. If T is a differential operator, the eigenvectors are commonly called eigenfunctions of the differential operator representing T. For example, differentiation itself is a linear transformation since

$\displaystyle {\frac {d}{dt}}(af+bg)=a{\frac {df}{dt}}+b{\frac {dg}{dt}}$

(f(t) and g(t) are differentiable functions, and a and b are constants).

Consider differentiation with respect to $t$. Its eigenfunctions h(t) obey the eigenvalue equation:

$\displaystyle {\frac {dh}{dt}}=\lambda h$,

where λ is the eigenvalue associated with the function. Such a function of time is constant if $\lambda =0$, grows proportionally to itself if $\lambda$ is positive, and decays proportionally to itself if $\lambda$ 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 $g(t)=\exp(\lambda t)$, 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 g 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 every 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 $g(t)=A\exp(\lambda t)$, where A is an arbitrary constant, the initial population at t=0.

Spectral theorem

Further information: spectral theorem

In its simplest version, the spectral theorem states that, under certain conditions, a linear transformation of a vector $\mathbf {v}$ can be expressed as a linear combination of the eigenvectors, in which the coefficient of each eigenvector is equal to the corresponding eigenvalue times the scalar product (or dot product) of the eigenvector with the vector $\mathbf {v}$. Mathematically, it can be written as:

${\mathcal {T}}(\mathbf {v} )=\lambda _{1}(\mathbf {v} _{1}\cdot \mathbf {v} )\mathbf {v} _{1}+\lambda _{2}(\mathbf {v} _{2}\cdot \mathbf {v} )\mathbf {v} _{2}+\cdots$

where $\mathbf {v} _{1},\mathbf {v} _{2},\dots$ and $\lambda _{1},\lambda _{2},\dots$ stand for the eigenvectors and eigenvalues of ${\mathcal {T}}$. The theorem is valid for all self-adjoint linear transformations (linear transformations given by real symmetric matrices and Hermitian matrices), and for the more general class of (complex) normal matrices.

If one defines the nth power of a transformation as the result of applying it n times in succession, one can also define polynomials of transformations. A more general version of the theorem is that any polynomial P of ${\mathcal {T}}$ is given by

$P({\mathcal {T}})(\mathbf {v} )=P(\lambda _{1})(\mathbf {v} _{1}\cdot \mathbf {v} )\mathbf {v} _{1}+P(\lambda _{2})(\mathbf {v} _{2}\cdot \mathbf {v} )\mathbf {v} _{2}+\cdots$

The theorem can be extended to other functions of transformations, such as analytic functions, the most general case being Borel functions.

Eigendecomposition

Main article: Eigendecomposition (matrix)

The spectral theorem for matrices can be stated as follows. Let $\mathbf {A}$ be a square ($n\times n$) matrix. Let $\mathbf {q} _{1}...\mathbf {q} _{k}$ be an eigenvector basis, i.e. an indexed set of k linearly independent eigenvectors, where k is the dimension of the space spanned by the eigenvectors of $\mathbf {A}$. If k=n, then $\mathbf {A}$ can be written

$\mathbf {A} =\mathbf {Q} \mathbf {\Lambda } \mathbf {Q} ^{-1}$

where $\mathbf {Q}$ is the square ($n\times n$) matrix whose i^th column is the basis eigenvector $\mathbf {q} _{i}$ of $\mathbf {A}$ and $\mathbf {\Lambda }$ is the diagonal matrix whose diagonal elements are the corresponding eigenvalues, i.e. $\Lambda _{ii}=\lambda _{i}$.

Infinite-dimensional spaces

If the vector space is an infinite dimensional Banach space, the notion of eigenvalues can be generalized to the concept of spectrum. The spectrum is the set of scalars λ for which $\left(T-\lambda \right)^{-1}$ is not defined; that is, such that $T-\lambda$ has no bounded inverse.

Clearly if λ is an eigenvalue of T, λ is in the spectrum of T. In general, the converse is not true. There are operators on Hilbert or Banach spaces which have no eigenvectors at all. This can be seen in the following example. The bilateral shift on the Hilbert space $\ell ^{2}(\mathbf {Z} )$ (the space of all sequences of scalars $\dots a_{-1},a_{0},a_{1},a_{2},\dots$ such that $\cdots +|a_{-1}|^{2}+|a_{0}|^{2}+|a_{1}|^{2}+|a_{2}|^{2}+\cdots$ converges) has no eigenvalue but has spectral values.

In infinite-dimensional spaces, the spectrum of a bounded operator is always nonempty. This is also true for an unbounded self adjoint operator. Via its spectral measures, the spectrum of any self adjoint operator, bounded or otherwise, can be decomposed into absolutely continuous, pure point, and singular parts. (See Decomposition of spectrum.)

Exponential functions are eigenfunctions of the derivative operator (the derivative of exponential functions are proportional to themself). Exponential growth and decay therefore provide examples of continuous spectra, as does the vibrating string example illustrated above. The hydrogen atom is an example where both types of spectra appear. The eigenfunctions of the hydrogen atom Hamiltonian are called eigenstates and are grouped into two categories. The bound states of the hydrogen atom correspond to the discrete part of the spectrum (they have a discrete set of eigenvalues which can be computed by Rydberg formula) while the ionization processes are described by the continuous part (the energy of the collision/ionization is not quantified).

Applications

Schrödinger equation

Fig. 4. The wavefunctions associated with 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 with eigenvalues interpreted as their energies (increasing downward: n=1,2,3,...) and angular momentum (increasing across: s, p, d,...). The illustration shows 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 nucleus, a proton.

An example of an eigenvalue equation where the transformation ${\mathcal {T}}$ is represented in terms of a differential operator is the time-independent Schrödinger equation in quantum mechanics:

$H\psi _{E}=E\psi _{E}\,$

where H, the Hamiltonian, is a second-order differential operator and $\psi _{E}$, the wavefunction, is one of its eigenfunctions corresponding to the eigenvalue E, interpreted as its energy.

However, in the case where one is interested only in the bound state solutions of the Schrödinger equation, one looks for $\psi _{E}$ within the space of square integrable functions. Since this space is a Hilbert space with a well-defined scalar product, one can introduce a basis set in which $\psi _{E}$ and H can be represented as a one-dimensional array and a matrix respectively. This allows one to represent the Schrödinger equation in a matrix form. (Fig. 4 presents the lowest eigenfunctions of the Hydrogen atom Hamiltonian.)

The Dirac notation is often used in this context. A vector, which represents a state of the system, in the Hilbert space of square integrable functions is represented by $|\Psi _{E}\rangle$. In this notation, the Schrödinger equation is:

$H|\Psi _{E}\rangle =E|\Psi _{E}\rangle$

where $|\Psi _{E}\rangle$ is an eigenstate of H. It is a self adjoint operator, the infinite dimensional analog of Hermitian matrices (see Observable). As in the matrix case, in the equation above $H|\Psi _{E}\rangle$ is understood to be the vector obtained by application of the transformation H to $|\Psi _{E}\rangle$.

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 on 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.

Geology and Glaciology: (Orientation Tensor)

In geology, especially in the study of glacial till, eigenvectors and eigenvalues are used as a method by which a mass of information of a clast fabric's constituents' orientation and dip can be summarized in a 3-D space by six numbers. In the field, a geologist may collect such data for hundreds or thousands of clasts in a soil sample, which can only be compared graphically such as in a Tri-Plot (Sneed and Folk) diagram , , or as a Stereonet on a Wulff Net . The output for the orientation tensor is in the three orthogonal (perpendicular) axes of space. Eigenvectors output from programs such as Stereo32 are in the order E1 > E2 > E3, with E1 being the primary orientation of clast orientation/dip, E2 being the secondary and E3 being the tertiary, in terms of strength. The clast orientation is defined as the Eigenvector, on a compass rose of 360°. Dip is measured as the Eigenvalue, the modulus of the tensor: this is valued from 0° (no dip) to 90° (vertical). Various values of E1, E2 and E3 mean different things, as can be seen in the book 'A Practical Guide to the Study of Glacial Sediments' by Benn & Evans, 2004 .

Factor analysis

In factor analysis, the eigenvectors of a covariance matrix or correlation matrix correspond to factors, and eigenvalues to the variance explained by these factors. 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 covariability among a number of observable random variables in terms of a smaller number of unobservable latent variables called factors. The observable random variables are modeled as linear combinations of the factors, plus unique variance terms. Eigenvalues are used in analysis used by Q-methodology software; factors with eigenvalues greater than 1.00 are considered significant, explaining an important amount of the variability in the data, while eigenvalues less than 1.00 are considered too weak, not explaining a significant portion of the data variability.

Fig. 5. Eigenfaces as examples of eigenvectors

Eigenfaces

In image processing, processed images of faces can be seen as vectors whose components are the brightnesses of each pixel. 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. In the facial recognition branch of biometrics, eigenfaces provide a means of applying data compression to faces for identification purposes. Research related to eigen vision systems determining hand gestures has also been made. More on determining sign language letters using eigen systems can be found here: http://www.geigel.com/signlanguage/index.php

Similar to this concept, eigenvoices concept is also developed which represents the general direction of variability in human pronunciations of a particular utterance, such as a word in a language. Based on a linear combination of such eigenvoices, a new voice pronunciation of the word can be constructed. These concepts have been found useful in automatic speech recognition systems, for speaker adaptation.

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.

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.

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, which is either T−A or 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. The kth principal eigenvector of a graph is defined as either the eigenvector corresponding to the kth largest eigenvalue of A, or the eigenvector corresponding to the kth smallest eigenvalue of the Laplacian. The first principal eigenvector of the graph is also referred to merely as the principal eigenvector.

The principal eigenvector is used to measure the centrality of its vertices. An example is Google's PageRank algorithm. The principal eigenvector of a modified adjacency matrix of the World Wide Web graph gives the page ranks as its components. This vector corresponds to the stationary distribution of the Markov chain represented by the row-normalized adjacency matrix; however, the adjacency matrix must first be modified to ensure a stationary distribution exists. The second principal eigenvector can be used to partition the graph into clusters, via spectral clustering.

See also

• Nonlinear eigenproblem
• Quadratic eigenvalue problem

References

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External links

• MIT Video Lecture on Eigenvalues and Eigenvectors at Google Video, from MIT OpenCourseWare
• ARPACK is a collection of FORTRAN subroutines for solving large scale (sparse) eigenproblems.
• IRBLEIGS, has MATLAB code with similar capabilities to ARPACK. (See this paper for a comparison between IRBLEIGS and ARPACK.)
• LAPACK is a collection of FORTRAN subroutines for solving dense linear algebra problems
• ALGLIB includes a partial port of the LAPACK to C++, C#, Delphi, etc.
• "Eigenvalue (of a matrix)". PlanetMath.
• MathWorld: Eigenvector
• Online calculator for Eigenvalues and Eigenvectors
• Online Matrix Calculator Calculates eigenvalues, eigenvectors and other decompositions of matrices online
• Vanderplaats Research and Development - Provides the SMS eigenvalue solver for Structural Finite Element. The solver is in the GENESIS program as well as other commercial programs. SMS can be easily use with MSC.Nastran or NX/Nastran via DMAPs.
• What are Eigen Values? from PhysLink.com's "Ask the Experts"
• Templates for the Solution of Algebraic Eigenvalue Problems Edited by Zhaojun Bai, James Demmel, Jack Dongarra, Axel Ruhe, and Henk van der Vorst (a guide to the numerical solution of eigenvalue problems)