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Vectors that map to their scalar multiples, and the associated scalars
In linear algebra, an eigenvector or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by , is the factor by which the eigenvector is scaled.
Geometrically, an eigenvector, corresponding to a real nonzero eigenvalue, points in a direction in which it is stretched by the transformation and the eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, the direction is reversed. Loosely speaking, in a multidimensional vector space, the eigenvector is not rotated.
If T is a linear transformation from a vector space V over a fieldF into itself and v is a nonzero vector in V, then v is an eigenvector of T if T(v) is a scalar multiple of v. This can be written as
where ? is a scalar in F, known as the eigenvalue, characteristic value, or characteristic root associated with v.
There is a direct correspondence between n-by-nsquare matrices and linear transformations from an n-dimensional vector space into itself, given any basis of the vector space. Hence, in a finite-dimensional vector space, it is equivalent to define eigenvalues and eigenvectors using either the language of matrices, or the language of linear transformations.
If V is finite-dimensional, the above equation is equivalent to
In essence, an eigenvector v of a linear transformation T is a nonzero vector that, when T is applied to it, does not change direction. Applying T to the eigenvector only scales the eigenvector by the scalar value ?, called an eigenvalue. This condition can be written as the equation
referred to as the eigenvalue equation or eigenequation. In general, ? may be any scalar. For example, ? may be negative, in which case the eigenvector reverses direction as part of the scaling, or it may be zero or complex.
In this shear mapping the red arrow changes direction, but the blue arrow does not. The blue arrow is an eigenvector of this shear mapping because it does not change direction, and since its length is unchanged, its eigenvalue is 1.
A 2×2 real and symmetric matrix represent a stretching and shearing of the plane. The eigenvectors of the matrix (red lines) are the two special directions such that every point on them will just slide on them.
The Mona Lisa example pictured here provides a simple illustration. Each point on the painting can be represented as a vector pointing from the center of the painting to that point. The linear transformation in this example is called a shear mapping. Points in the top half are moved to the right, and points in the bottom half are moved to the left, proportional to how far they are from the horizontal axis that goes through the middle of the painting. The vectors pointing to each point in the original image are therefore tilted right or left, and made longer or shorter by the transformation. Points along the horizontal axis do not move at all when this transformation is applied. Therefore, any vector that points directly to the right or left with no vertical component is an eigenvector of this transformation, because the mapping does not change its direction. Moreover, these eigenvectors all have an eigenvalue equal to one, because the mapping does not change their length either.
Linear transformations can take many different forms, mapping vectors in a variety of vector spaces, so the eigenvectors can also take many forms. For example, the linear transformation could be a differential operator like , in which case the eigenvectors are functions called eigenfunctions that are scaled by that differential operator, such as
Alternatively, the linear transformation could take the form of an n by n matrix, in which case the eigenvectors are n by 1 matrices. If the linear transformation is expressed in the form of an n by n matrix A, then the eigenvalue equation for a linear transformation above can be rewritten as the matrix multiplication
At the start of the 20th century, David Hilbert studied the eigenvalues of integral operators by viewing the operators as infinite matrices. He was the first to use the German word eigen, which means "own", to denote eigenvalues and eigenvectors in 1904,[c] though he may have been following a related usage by Hermann von Helmholtz. For some time, the standard term in English was "proper value", but the more distinctive term "eigenvalue" is the standard today.
Eigenvalues and eigenvectors are often introduced to students in the context of linear algebra courses focused on matrices.
Furthermore, linear transformations over a finite-dimensional vector space can be represented using matrices, which is especially common in numerical and computational applications.
Matrix A acts by stretching the vector x, not changing its direction, so x is an eigenvector of A.
Consider n-dimensional vectors that are formed as a list of n scalars, such as the three-dimensional vectors
Equation (2) has a nonzero solution vif and only if the determinant of the matrix is zero. Therefore, the eigenvalues of A are values of ? that satisfy the equation
Using Leibniz' rule for the determinant, the left-hand side of Equation (3) is a polynomial function of the variable ? and the degree of this polynomial is n, the order of the matrix A. Its coefficients depend on the entries of A, except that its term of degree n is always (-1)n?n. This polynomial is called the characteristic polynomial of A. Equation (3) is called the characteristic equation or the secular equation of A.
where each ?i may be real but in general is a complex number. The numbers ?1, ?2, ..., ?n, which may not all have distinct values, are roots of the polynomial and are the eigenvalues of A.
As a brief example, which is described in more detail in the examples section later, consider the matrix
Taking the determinant of , the characteristic polynomial of A is
Setting the characteristic polynomial equal to zero, it has roots at and , which are the two eigenvalues of A. The eigenvectors corresponding to each eigenvalue can be found by solving for the components of v in the equation In this example, the eigenvectors are any nonzero scalar multiples of
If the entries of the matrix A are all real numbers, then the coefficients of the characteristic polynomial will also be real numbers, but the eigenvalues may still have nonzero imaginary parts. The entries of the corresponding eigenvectors therefore may also have nonzero imaginary parts. Similarly, the eigenvalues may be irrational numbers even if all the entries of A are rational numbers or even if they are all integers. However, if the entries of A are all algebraic numbers, which include the rationals, the eigenvalues are complex algebraic numbers.
The non-real roots of a real polynomial with real coefficients can be grouped into pairs of complex conjugates, namely with the two members of each pair having imaginary parts that differ only in sign and the same real part. If the degree is odd, then by the intermediate value theorem at least one of the roots is real. Therefore, any real matrix with odd order has at least one real eigenvalue, whereas a real matrix with even order may not have any real eigenvalues. The eigenvectors associated with these complex eigenvalues are also complex and also appear in complex conjugate pairs.
Let ?i be an eigenvalue of an n by n matrix A. The algebraic multiplicity?A(?i) of the eigenvalue is its multiplicity as a root of the characteristic polynomial, that is, the largest integer k such that (? - ?i)kdivides evenly that polynomial.
Suppose a matrix A has dimension n and d n distinct eigenvalues. Whereas Equation (4) factors the characteristic polynomial of A into the product of n linear terms with some terms potentially repeating, the characteristic polynomial can instead be written as the product of d terms each corresponding to a distinct eigenvalue and raised to the power of the algebraic multiplicity,
If d = n then the right-hand side is the product of n linear terms and this is the same as Equation (4). The size of each eigenvalue's algebraic multiplicity is related to the dimension n as
If ?A(?i) = 1, then ?i is said to be a simple eigenvalue. If ?A(?i) equals the geometric multiplicity of ?i, ?A(?i), defined in the next section, then ?i is said to be a semisimple eigenvalue.
Eigenspaces, geometric multiplicity, and the eigenbasis for matrices
Given a particular eigenvalue ? of the n by n matrix A, define the setE to be all vectors v that satisfy Equation (2),
On one hand, this set is precisely the kernel or nullspace of the matrix (A - ?I). On the other hand, by definition, any nonzero vector that satisfies this condition is an eigenvector of A associated with ?. So, the set E is the union of the zero vector with the set of all eigenvectors of A associated with ?, and E equals the nullspace of (A - ?I). E is called the eigenspace or characteristic space of A associated with ?. In general ? is a complex number and the eigenvectors are complex n by 1 matrices. A property of the nullspace is that it is a linear subspace, so E is a linear subspace of Cn.
Because the eigenspace E is a linear subspace, it is closed under addition. That is, if two vectors u and v belong to the set E, written , then or equivalently . This can be checked using the distributive property of matrix multiplication. Similarly, because E is a linear subspace, it is closed under scalar multiplication. That is, if and ? is a complex number, or equivalently . This can be checked by noting that multiplication of complex matrices by complex numbers is commutative. As long as u + v and ?v are not zero, they are also eigenvectors of A associated with ?.
The dimension of the eigenspace E associated with ?, or equivalently the maximum number of linearly independent eigenvectors associated with ?, is referred to as the eigenvalue's geometric multiplicity?A(?). Because E is also the nullspace of (A - ?I), the geometric multiplicity of ? is the dimension of the nullspace of (A - ?I), also called the nullity of (A - ?I), which relates to the dimension and rank of (A - ?I) as
Because of the definition of eigenvalues and eigenvectors, an eigenvalue's geometric multiplicity must be at least one, that is, each eigenvalue has at least one associated eigenvector. Furthermore, an eigenvalue's geometric multiplicity cannot exceed its algebraic multiplicity. Additionally, recall that an eigenvalue's algebraic multiplicity cannot exceed n.
To prove the inequality , consider how the definition of geometric multiplicity implies the existence of orthonormal eigenvectors , such that . We can therefore find a (unitary) matrix whose first columns are these eigenvectors, and whose remaining columns can be any orthonormal set of vectors orthogonal to these eigenvectors of . Then has full rank and is therefore invertible, and with a matrix whose top left block is the diagonal matrix . This implies that . In other words, is similar to , which implies that . But from the definition of we know that contains a factor , which means that the algebraic multiplicity of must satisfy .
Suppose has distinct eigenvalues , where the geometric multiplicity of is . The total geometric multiplicity of ,
is the dimension of the sum of all the eigenspaces of 's eigenvalues, or equivalently the maximum number of linearly independent eigenvectors of . If , then
The direct sum of the eigenspaces of all of 's eigenvalues is the entire vector space .
A basis of can be formed from linearly independent eigenvectors of ; such a basis is called an eigenbasis
Any vector in can be written as a linear combination of eigenvectors of .
Additional properties of eigenvalues
Let be an arbitrary matrix of complex numbers with eigenvalues . Each eigenvalue appears times in this list, where is the eigenvalue's algebraic multiplicity. The following are properties of this matrix and its eigenvalues:
The trace of , defined as the sum of its diagonal elements, is also the sum of all eigenvalues,
The eigenvalues of the th power of ; i.e., the eigenvalues of , for any positive integer , are .
The matrix is invertible if and only if every eigenvalue is nonzero.
If is invertible, then the eigenvalues of are and each eigenvalue's geometric multiplicity coincides. Moreover, since the characteristic polynomial of the inverse is the reciprocal polynomial of the original, the eigenvalues share the same algebraic multiplicity.
If is not only Hermitian but also positive-definite, positive-semidefinite, negative-definite, or negative-semidefinite, then every eigenvalue is positive, non-negative, negative, or non-positive, respectively.
If is unitary, every eigenvalue has absolute value .
if is a matrix and are its eigenvalues, then the eigenvalues of matrix (where is the identity matrix) are . Moreover, if , the eigenvalues of are . More generally, for a polynomial the eigenvalues of matrix are .
Left and right eigenvectors
Many disciplines traditionally represent vectors as matrices with a single column rather than as matrices with a single row. For that reason, the word "eigenvector" in the context of matrices almost always refers to a right eigenvector, namely a column vector that right multiplies the matrix in the defining equation, Equation (1),
The eigenvalue and eigenvector problem can also be defined for row vectors that left multiply matrix . In this formulation, the defining equation is
where is a scalar and is a matrix. Any row vector satisfying this equation is called a left eigenvector of and is its associated eigenvalue. Taking the transpose of this equation,
Comparing this equation to Equation (1), it follows immediately that a left eigenvector of is the same as the transpose of a right eigenvector of , with the same eigenvalue. Furthermore, since the characteristic polynomial of is the same as the characteristic polynomial of , the eigenvalues of the left eigenvectors of are the same as the eigenvalues of the right eigenvectors of .
Diagonalization and the eigendecomposition
Suppose the eigenvectors of A form a basis, or equivalently A has n linearly independent eigenvectors v1, v2, ..., vn with associated eigenvalues ?1, ?2, ..., ?n. The eigenvalues need not be distinct. Define a square matrix Q whose columns are the n linearly independent eigenvectors of A,
Since each column of Q is an eigenvector of A, right multiplying A by Q scales each column of Q by its associated eigenvalue,
With this in mind, define a diagonal matrix ? where each diagonal element ?ii is the eigenvalue associated with the ith column of Q. Then
Because the columns of Q are linearly independent, Q is invertible. Right multiplying both sides of the equation by Q-1,
or by instead left multiplying both sides by Q-1,
A can therefore be decomposed into a matrix composed of its eigenvectors, a diagonal matrix with its eigenvalues along the diagonal, and the inverse of the matrix of eigenvectors. This is called the eigendecomposition and it is a similarity transformation. Such a matrix A is said to be similar to the diagonal matrix ? or diagonalizable. The matrix Q is the change of basis matrix of the similarity transformation. Essentially, the matrices A and ? represent the same linear transformation expressed in two different bases. The eigenvectors are used as the basis when representing the linear transformation as ?.
Conversely, suppose a matrix A is diagonalizable. Let P be a non-singular square matrix such that P-1AP is some diagonal matrix D. Left multiplying both by P, . Each column of P must therefore be an eigenvector of A whose eigenvalue is the corresponding diagonal element of D. Since the columns of P must be linearly independent for P to be invertible, there exist n linearly independent eigenvectors of A. It then follows that the eigenvectors of A form a basis if and only if A is diagonalizable.
In the Hermitian case, eigenvalues can be given a variational characterization. The largest eigenvalue of is the maximum value of the quadratic form. A value of that realizes that maximum, is an eigenvector.
Two-dimensional matrix example
The transformation matrix A = preserves the direction of purple vectors parallel to v?=1 = [1 -1]T and blue vectors parallel to v?=3 = [1 1]T. The red vectors are not parallel to either eigenvector, so, their directions are changed by the transformation. The lengths of the purple vectors are unchanged after the transformation (due to their eigenvalue of 1), while blue vectors are three times the length of the original (due to their eigenvalue of 3). See also: An extended version, showing all four quadrants.
Consider the matrix
The figure on the right shows the effect of this transformation on point coordinates in the plane. The eigenvectors v of this transformation satisfy Equation (1), and the values of ? for which the determinant of the matrix (A - ?I) equals zero are the eigenvalues.
Taking the determinant to find characteristic polynomial of A,
Setting the characteristic polynomial equal to zero, it has roots at and , which are the two eigenvalues of A.
For the real eigenvalue ?1 = 1, any vector with three equal nonzero entries is an eigenvector. For example,
For the complex conjugate pair of imaginary eigenvalues,
Therefore, the other two eigenvectors of A are complex and are and with eigenvalues ?2 and ?3, respectively. The two complex eigenvectors also appear in a complex conjugate pair,
Diagonal matrix example
Matrices with entries only along the main diagonal are called diagonal matrices. The eigenvalues of a diagonal matrix are the diagonal elements themselves. Consider the matrix
The characteristic polynomial of A is
which has the roots , , and . These roots are the diagonal elements as well as the eigenvalues of A.
Each diagonal element corresponds to an eigenvector whose only nonzero component is in the same row as that diagonal element. In the example, the eigenvalues correspond to the eigenvectors,
respectively, as well as scalar multiples of these vectors.
Triangular matrix example
A matrix whose elements above the main diagonal are all zero is called a lower triangular matrix, while a matrix whose elements below the main diagonal are all zero is called an upper triangular matrix. As with diagonal matrices, the eigenvalues of triangular matrices are the elements of the main diagonal.
Consider the lower triangular matrix,
The characteristic polynomial of A is
which has the roots , , and . These roots are the diagonal elements as well as the eigenvalues of A.
These eigenvalues correspond to the eigenvectors,
respectively, as well as scalar multiples of these vectors.
Matrix with repeated eigenvalues example
As in the previous example, the lower triangular matrix
has a characteristic polynomial that is the product of its diagonal elements,
The roots of this polynomial, and hence the eigenvalues, are 2 and 3. The algebraic multiplicity of each eigenvalue is 2; in other words they are both double roots. The sum of the algebraic multiplicities of all distinct eigenvalues is ?A = 4 = n, the order of the characteristic polynomial and the dimension of A.
On the other hand, the geometric multiplicity of the eigenvalue 2 is only 1, because its eigenspace is spanned by just one vector and is therefore 1-dimensional. Similarly, the geometric multiplicity of the eigenvalue 3 is 1 because its eigenspace is spanned by just one vector . The total geometric multiplicity ?A is 2, which is the smallest it could be for a matrix with two distinct eigenvalues. Geometric multiplicities are defined in a later section.
For a Hermitian matrix, the norm squared of the jth component of a normalized eigenvector can be calculated using only the matrix eigenvalues and the eigenvalues of the corresponding minor matrix,
Eigenvalues and eigenfunctions of differential operators
The definitions of eigenvalue and eigenvectors of a linear transformation T remains valid even if the underlying vector space is an infinite-dimensional Hilbert or Banach space. A widely used class of linear transformations acting on infinite-dimensional spaces are the differential operators on function spaces. Let D be a linear differential operator on the space C? of infinitely differentiable real functions of a real argument t. The eigenvalue equation for D is the differential equation
The functions that satisfy this equation are eigenvectors of D and are commonly called eigenfunctions.
Derivative operator example
Consider the derivative operator with eigenvalue equation
This differential equation can be solved by multiplying both sides by dt/f(t) and integrating. Its solution, the exponential function
is the eigenfunction of the derivative operator. In this case the eigenfunction is itself a function of its associated eigenvalue. In particular, for ? = 0 the eigenfunction f(t) is a constant.
The concept of eigenvalues and eigenvectors extends naturally to arbitrary linear transformations on arbitrary vector spaces. Let V be any vector space over some fieldK of scalars, and let T be a linear transformation mapping V into V,
We say that a nonzero vector v ? V is an eigenvector of T if and only if there exists a scalar ? ? K such that
This equation is called the eigenvalue equation for T, and the scalar ? is the eigenvalue of T corresponding to the eigenvector v. T(v) is the result of applying the transformation T to the vector v, while ?v is the product of the scalar ? with v.
Eigenspaces, geometric multiplicity, and the eigenbasis
Given an eigenvalue ?, consider the set
which is the union of the zero vector with the set of all eigenvectors associated with ?. E is called the eigenspace or characteristic space of T associated with ?.
By definition of a linear transformation,
for x, y ? V and ? ? K. Therefore, if u and v are eigenvectors of T associated with eigenvalue ?, namely u, v ? E, then
So, both u + v and ?v are either zero or eigenvectors of T associated with ?, namely u + v, ?v ? E, and E is closed under addition and scalar multiplication. The eigenspace E associated with ? is therefore a linear subspace of V.
If that subspace has dimension 1, it is sometimes called an eigenline.
The geometric multiplicity?T(?) of an eigenvalue ? is the dimension of the eigenspace associated with ?, i.e., the maximum number of linearly independent eigenvectors associated with that eigenvalue. By the definition of eigenvalues and eigenvectors, ?T(?) >= 1 because every eigenvalue has at least one eigenvector.
The eigenspaces of T always form a direct sum. As a consequence, eigenvectors of different eigenvalues are always linearly independent. Therefore, the sum of the dimensions of the eigenspaces cannot exceed the dimension n of the vector space on which T operates, and there cannot be more than n distinct eigenvalues.[e]
Any subspace spanned by eigenvectors of T is an invariant subspace of T, and the restriction of T to such a subspace is diagonalizable. Moreover, if the entire vector space V can be spanned by the eigenvectors of T, or equivalently if the direct sum of the eigenspaces associated with all the eigenvalues of T is the entire vector space V, then a basis of V called an eigenbasis can be formed from linearly independent eigenvectors of T. When T admits an eigenbasis, T is diagonalizable.
Zero vector as an eigenvector
While the definition of an eigenvector used in this article excludes the zero vector, it is possible to define eigenvalues and eigenvectors such that the zero vector is an eigenvector.
Consider again the eigenvalue equation, Equation (5). Define an eigenvalue to be any scalar ? ? K such that there exists a nonzero vector v ? V satisfying Equation (5). It is important that this version of the definition of an eigenvalue specify that the vector be nonzero, otherwise by this definition the zero vector would allow any scalar in K to be an eigenvalue. Define an eigenvectorv associated with the eigenvalue ? to be any vector that, given ?, satisfies Equation (5). Given the eigenvalue, the zero vector is among the vectors that satisfy Equation (5), so the zero vector is included among the eigenvectors by this alternate definition.
If ? is an eigenvalue of T, then the operator (T - ?I) is not one-to-one, and therefore its inverse (T - ?I)-1 does not exist. The converse is true for finite-dimensional vector spaces, but not for infinite-dimensional vector spaces. In general, the operator (T - ?I) may not have an inverse even if ? is not an eigenvalue.
For this reason, in functional analysis eigenvalues can be generalized to the spectrum of a linear operatorT as the set of all scalars ? for which the operator (T - ?I) has no bounded inverse. The spectrum of an operator always contains all its eigenvalues but is not limited to them.
The solution of this equation for x in terms of t is found by using its characteristic equation
which can be found by stacking into matrix form a set of equations consisting of the above difference equation and the k - 1 equations giving a k-dimensional system of the first order in the stacked variable vector in terms of its once-lagged value, and taking the characteristic equation of this system's matrix. This equation gives k characteristic roots for use in the solution equation
The calculation of eigenvalues and eigenvectors is a topic where theory, as presented in elementary linear algebra textbooks, is often very far from practice.
The classical method is to first find the eigenvalues, and then calculate the eigenvectors for each eigenvalue. It is in several ways poorly suited for non-exact arithmetics such as floating-point.
The eigenvalues of a matrix can be determined by finding the roots of the characteristic polynomial. This is easy for matrices, but the difficulty increases rapidly with the size of the matrix.
In theory, the coefficients of the characteristic polynomial can be computed exactly, since they are sums of products of matrix elements; and there are algorithms that can find all the roots of a polynomial of arbitrary degree to any required accuracy. However, this approach is not viable in practice because the coefficients would be contaminated by unavoidable round-off errors, and the roots of a polynomial can be an extremely sensitive function of the coefficients (as exemplified by Wilkinson's polynomial). Even for matrices whose elements are integers the calculation becomes nontrivial, because the sums are very long; the constant term is the determinant, which for an is a sum of different products.[f]
Explicit algebraic formulas for the roots of a polynomial exist only if the degree is 4 or less. According to the Abel-Ruffini theorem there is no general, explicit and exact algebraic formula for the roots of a polynomial with degree 5 or more. (Generality matters because any polynomial with degree is the characteristic polynomial of some companion matrix of order .) Therefore, for matrices of order 5 or more, the eigenvalues and eigenvectors cannot be obtained by an explicit algebraic formula, and must therefore be computed by approximate numerical methods. Even the exact formula for the roots of a degree 3 polynomial is numerically impractical.
Once the (exact) value of an eigenvalue is known, the corresponding eigenvectors can be found by finding nonzero solutions of the eigenvalue equation, that becomes a system of linear equations with known coefficients. For example, once it is known that 6 is an eigenvalue of the matrix
we can find its eigenvectors by solving the equation , that is
Both equations reduce to the single linear equation . Therefore, any vector of the form , for any nonzero real number , is an eigenvector of with eigenvalue .
The matrix above has another eigenvalue . A similar calculation shows that the corresponding eigenvectors are the nonzero solutions of , that is, any vector of the form , for any nonzero real number .
Simple iterative methods
The converse approach, of first seeking the eigenvectors and then determining each eigenvalue from its eigenvector, turns out to be far more tractable for computers. The easiest algorithm here consists of picking an arbitrary starting vector and then repeatedly multiplying it with the matrix (optionally normalizing the vector to keep its elements of reasonable size); this makes the vector converge towards an eigenvector. A variation is to instead multiply the vector by this causes it to converge to an eigenvector of the eigenvalue closest to
If is (a good approximation of) an eigenvector of , then the corresponding eigenvalue can be computed as
Most numeric methods that compute the eigenvalues of a matrix also determine a set of corresponding eigenvectors as a by-product of the computation, although sometimes implementors choose to discard the eigenvector information as soon as it is no longer needed.
Eigenvalues of geometric transformations
The following table presents some example transformations in the plane along with their 2×2 matrices, eigenvalues, and eigenvectors.
The characteristic equation for a rotation is a quadratic equation with discriminant, which is a negative number whenever ? is not an integer multiple of 180°. Therefore, except for these special cases, the two eigenvalues are complex numbers, ; and all eigenvectors have non-real entries. Indeed, except for those special cases, a rotation changes the direction of every nonzero vector in the plane.
A linear transformation that takes a square to a rectangle of the same area (a squeeze mapping) has reciprocal eigenvalues.
However, in the case where one is interested only in the bound state solutions of the Schrödinger equation, one looks for 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 and can be represented as a one-dimensional array (i.e., a vector) and a matrix respectively. This allows one to represent the Schrödinger equation in a matrix form.
The bra-ket 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 . In this notation, the Schrödinger equation is:
where is an eigenstate of and represents the eigenvalue. is an observableself-adjoint operator, the infinite-dimensional analog of Hermitian matrices. As in the matrix case, in the equation above is understood to be the vector obtained by application of the transformation to .
Light, acoustic waves, and microwaves are randomly scattered numerous times when traversing a static disordered system. Even though multiple scattering repeatedly randomizes the waves, ultimately coherent wave transport through the system is a deterministic process which can be described by a field transmission matrix . The eigenvectors of the transmission operator form a set of disorder-specific input wavefronts which enable waves to couple into the disordered system's eigenchannels: the independent pathways waves can travel through the system. The eigenvalues, , of correspond to the intensity transmittance associated with each eigenchannel. One of the remarkable properties of the transmission operator of diffusive systems is their bimodal eigenvalue distribution with and . Furthermore, one of the striking properties of open eigenchannels, beyond the perfect transmittance, is the statistically robust spatial profile of the eigenchannels.
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. The three eigenvectors are ordered by their eigenvalues ; then is the primary orientation/dip of clast, is the secondary and is the tertiary, in terms of strength. The clast orientation is defined as the direction of 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). The relative values of , , and are dictated by the nature of the sediment's fabric. If , the fabric is said to be isotropic. If , the fabric is said to be planar. If , the fabric is said to be linear.
Principal component analysis
PCA of the multivariate Gaussian distribution centered at with a standard deviation of 3 in roughly the direction and of 1 in the orthogonal direction. The vectors shown are unit eigenvectors of the (symmetric, positive-semidefinite) covariance matrix scaled by the square root of the corresponding eigenvalue. Just as in the one-dimensional case, the square root is taken because the standard deviation is more readily visualized than the variance.
Mode shape of a tuning fork at eigenfrequency 440.09 Hz
Eigenvalue problems occur naturally in the vibration analysis of mechanical structures with many degrees of freedom. The eigenvalues are the natural frequencies (or eigenfrequencies) of vibration, and the eigenvectors are the shapes of these vibrational modes. In particular, undamped vibration is governed by
that is, acceleration is proportional to position (i.e., we expect to be sinusoidal in time).
where is the eigenvalue and is the (imaginary) angular frequency. The principal vibration modes are different from the principal compliance modes, which are the eigenvectors of alone. Furthermore, damped vibration, governed by
This can be reduced to a generalized eigenvalue problem by algebraic manipulation at the cost of solving a larger system.
The orthogonality properties of the eigenvectors allows decoupling of the differential equations so that the system can be represented as linear summation of the eigenvectors. The eigenvalue problem of complex structures is often solved using finite element analysis, but neatly generalize the solution to scalar-valued vibration problems.
Similar to this concept, eigenvoices represent 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.
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.
In spectral graph theory, an eigenvalue of a graph is defined as an eigenvalue of the graph's adjacency matrix, or (increasingly) of the graph's Laplacian matrix due to its discrete Laplace operator, which is either (sometimes called the combinatorial Laplacian) or (sometimes called the normalized Laplacian), where is a diagonal matrix with equal to the degree of vertex , and in , the th diagonal entry is . The th principal eigenvector of a graph is defined as either the eigenvector corresponding to the th largest or th 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 smallest eigenvector can be used to partition the graph into clusters, via spectral clustering. Other methods are also available for clustering.
Basic reproduction number
The basic reproduction number () is a fundamental number in the study of how infectious diseases spread. If one infectious person is put into a population of completely susceptible people, then is the average number of people that one typical infectious person will infect. The generation time of an infection is the time, , from one person becoming infected to the next person becoming infected. In a heterogeneous population, the next generation matrix defines how many people in the population will become infected after time has passed. is then the largest eigenvalue of the next generation matrix.
In 1751, Leonhard Euler proved that any body has a principal axis of rotation: Leonhard Euler (presented: October 1751; published: 1760) "Du mouvement d'un corps solide quelconque lorsqu'il tourne autour d'un axe mobile" (On the movement of any solid body while it rotates around a moving axis), Histoire de l'Académie royale des sciences et des belles lettres de Berlin, pp. 176-227. On p. 212, Euler proves that any body contains a principal axis of rotation: "Théorem. 44. De quelque figure que soit le corps, on y peut toujours assigner un tel axe, qui passe par son centre de gravité, autour duquel le corps peut tourner librement & d'un mouvement uniforme." (Theorem. 44. Whatever be the shape of the body, one can always assign to it such an axis, which passes through its center of gravity, around which it can rotate freely and with a uniform motion.)
In 1755, Johann Andreas Segner proved that any body has three principal axes of rotation: Johann Andreas Segner, Specimen theoriae turbinum [Essay on the theory of tops (i.e., rotating bodies)] ( Halle ("Halae"), (Germany): Gebauer, 1755). (https://books.google.com/books?id=29 p. xxviiii ), Segner derives a third-degree equation in t, which proves that a body has three principal axes of rotation. He then states (on the same page): "Non autem repugnat tres esse eiusmodi positiones plani HM, quia in aequatione cubica radices tres esse possunt, et tres tangentis t valores." (However, it is not inconsistent [that there] be three such positions of the plane HM, because in cubic equations, [there] can be three roots, and three values of the tangent t.)
The relevant passage of Segner's work was discussed briefly by Arthur Cayley. See: A. Cayley (1862) "Report on the progress of the solution of certain special problems of dynamics," Report of the Thirty-second meeting of the British Association for the Advancement of Science; held at Cambridge in October 1862, 32: 184-252; see especially pp. 225-226.
^Kline 1972, pp. 807-808 Augustin Cauchy (1839) "Mémoire sur l'intégration des équations linéaires" (Memoir on the integration of linear equations), Comptes rendus, 8: 827-830, 845-865, 889-907, 931-937. From p. 827:"On sait d'ailleurs qu'en suivant la méthode de Lagrange, on obtient pour valeur générale de la variable prinicipale une fonction dans laquelle entrent avec la variable principale les racines d'une certaine équation que j'appellerai l'équation caractéristique, le degré de cette équation étant précisément l'order de l'équation différentielle qu'il s'agit d'intégrer." (One knows, moreover, that by following Lagrange's method, one obtains for the general value of the principal variable a function in which there appear, together with the principal variable, the roots of a certain equation that I will call the "characteristic equation", the degree of this equation being precisely the order of the differential equation that must be integrated.)
David Hilbert (1904) "Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen. (Erste Mitteilung)" (Fundamentals of a general theory of linear integral equations. (First report)), Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (News of the Philosophical Society at Göttingen, mathematical-physical section), pp. 49-91. From p. 51:"Insbesondere in dieser ersten Mitteilung gelange ich zu Formeln, die die Entwickelung einer willkürlichen Funktion nach gewissen ausgezeichneten Funktionen, die ich 'Eigenfunktionen' nenne, liefern: ..." (In particular, in this first report I arrive at formulas that provide the [series] development of an arbitrary function in terms of some distinctive functions, which I call eigenfunctions: ... ) Later on the same page: "Dieser Erfolg ist wesentlich durch den Umstand bedingt, daß ich nicht, wie es bisher geschah, in erster Linie auf den Beweis für die Existenz der Eigenwerte ausgehe, ... " (This success is mainly attributable to the fact that I do not, as it has happened until now, first of all aim at a proof of the existence of eigenvalues, ... )
Denton, Peter B.; Parke, Stephen J.; Tao, Terence; Zhang, Xining (2021). "Eigenvectors from eigenvalues: A survey of a basic identity in linear algebra". Bulletin of the American Mathematical Society: 1. arXiv:1908.03795. doi:10.1090/bull/1722. S2CID213918682.
Diekmann O, Heesterbeek JA, Metz JA (1990), "On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations", Journal of Mathematical Biology, 28 (4): 365-382, doi:10.1007/BF00178324, hdl:1874/8051, PMID2117040, S2CID22275430
Kublanovskaya, Vera N. (1961), "On some algorithms for the solution of the complete eigenvalue problem", USSR Computational Mathematics and Mathematical Physics, 3: 637-657. Also published in: "? ? ? " [On certain algorithms for the solution of the complete eigenvalue problem], ? ? (Journal of Computational Mathematics and Mathematical Physics) (in Russian), 1 (4): 555-570, 1961
Matrix Eigenvectors Calculator from Symbolab (Click on the bottom right button of the 2x12 grid to select a matrix size. Select an size (for a square matrix), then fill out the entries numerically and click on the Go button. It can accept complex numbers as well.)