Eigenvalues and eigenvectors

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 formula_1, 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.
Formal definition.
If is a linear transformation from a vector space over a field into itself and is a nonzero vector in , then is an eigenvector of if is a scalar multiple of . This can be written as
where is a scalar in , known as the eigenvalue, characteristic value, or characteristic root associated with .
There is a direct correspondence between "n"-by-"n" square 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 is finite-dimensional, the above equation is equivalent to
where is the matrix representation of and is the coordinate vector of .
Overview.
Eigenvalues and eigenvectors feature prominently in the analysis of linear transformations. The prefix "eigen-" is adopted from the German word "eigen" (cognate with the English word "own") for "proper", "characteristic", "own". Originally used to study principal axes of the rotational motion of rigid bodies, eigenvalues and eigenvectors have a wide range of applications, for example in stability analysis, vibration analysis, atomic orbitals, facial recognition, and matrix diagonalization.
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 "λ", called an eigenvalue. This condition can be written as the equation
referred to as the eigenvalue equation or eigenequation. In general, "λ" may be any scalar. For example, "λ" may be negative, in which case the eigenvector reverses direction as part of the scaling, or it may be zero or complex.
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 formula_5, 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
where the eigenvector "v" is an "n" by 1 matrix. For a matrix, eigenvalues and eigenvectors can be used to decompose the matrix—for example by diagonalizing it.
Eigenvalues and eigenvectors give rise to many closely related mathematical concepts, and the prefix "eigen-" is applied liberally when naming them:
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.
In the 18th century, Leonhard Euler studied the rotational motion of a rigid body, and discovered the importance of the principal axes. Joseph-Louis Lagrange realized that the principal axes are the eigenvectors of the inertia matrix.
In the early 19th century, Augustin-Louis 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".
Later, Joseph Fourier used the work of Lagrange and Pierre-Simon Laplace to solve the heat equation by separation of variables in his famous 1822 book "Théorie analytique de la chaleur". Charles-François Sturm developed Fourier's ideas further, and brought them to the attention of Cauchy, who combined them with his own ideas and arrived at the fact that real symmetric matrices have real eigenvalues. This was extended by Charles Hermite in 1855 to what are now called Hermitian matrices.
Around the same time, Francesco Brioschi proved that the eigenvalues of orthogonal matrices lie on the unit circle, and Alfred Clebsch found the corresponding result for skew-symmetric matrices. Finally, Karl Weierstrass clarified an important aspect in the stability theory started by Laplace, by realizing that defective matrices can cause instability.
In the meantime, Joseph Liouville studied eigenvalue problems similar to those of 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, 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, 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.
The first numerical algorithm for computing eigenvalues and eigenvectors appeared in 1929, when Richard von Mises published the power method. One of the most popular methods today, the QR algorithm, was proposed independently by John G. F. Francis and Vera Kublanovskaya in 1961.
Eigenvalues and eigenvectors of matrices.
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.
Consider -dimensional vectors that are formed as a list of scalars, such as the three-dimensional vectors
These vectors are said to be scalar multiples of each other, or parallel or collinear, if there is a scalar such that
In this case formula_10.
Now consider the linear transformation of -dimensional vectors defined by an by matrix ,
or
where, for each row,
If it occurs that and are scalar multiples, that is if
then is an eigenvector of the linear transformation and the scale factor is the eigenvalue corresponding to that eigenvector. Equation () is the eigenvalue equation for the matrix .
Equation () can be stated equivalently as
where is the by identity matrix and 0 is the zero vector.
Eigenvalues and the characteristic polynomial.
Equation () has a nonzero solution "v" if and only if the determinant of the matrix is zero. Therefore, the eigenvalues of "A" are values of "λ" that satisfy the equation
Using Leibniz' rule for the determinant, the left-hand side of Equation () is a polynomial function of the variable "λ" 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""λ""n". This polynomial is called the "characteristic polynomial" of "A". Equation () is called the "characteristic equation" or the "secular equation" of "A".
The fundamental theorem of algebra implies that the characteristic polynomial of an "n"-by-"n" matrix "A", being a polynomial of degree "n", can be factored into the product of "n" linear terms,
where each "λ""i" may be real but in general is a complex number. The numbers "λ"1, "λ"2, …, "λ""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.
Algebraic multiplicity.
Let "λ""i" be an eigenvalue of an "n" by "n" matrix "A". The algebraic multiplicity "μ""A"("λ""i") of the eigenvalue is its multiplicity as a root of the characteristic polynomial, that is, the largest integer "k" such that ("λ" − "λ""i")"k" divides evenly that polynomial.
Suppose a matrix "A" has dimension "n" and "d" ≤ "n" distinct eigenvalues. Whereas Equation () 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 (). The size of each eigenvalue's algebraic multiplicity is related to the dimension "n" as
If "μ""A"("λ""i") = 1, then "λ""i" is said to be a "simple eigenvalue". If "μ""A"("λ""i") equals the geometric multiplicity of "λ""i", "γ""A"("λ""i"), defined in the next section, then "λ""i" is said to be a "semisimple eigenvalue".
Eigenspaces, geometric multiplicity, and the eigenbasis for matrices.
Given a particular eigenvalue "λ" of the "n" by "n" matrix "A", define the set "E" to be all vectors v that satisfy Equation (),
On one hand, this set is precisely the kernel or nullspace of the matrix ("A" − "λI"). On the other hand, by definition, any nonzero vector that satisfies this condition is an eigenvector of "A" associated with "λ". So, the set "E" is the union of the zero vector with the set of all eigenvectors of "A" associated with "λ", and "E" equals the nullspace of ("A" − "λI"). "E" is called the eigenspace or characteristic space of "A" associated with "λ". In general "λ" 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 ℂ"n".
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 "α" 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 "α"v are not zero, they are also eigenvectors of "A" associated with "λ".
The dimension of the eigenspace "E" associated with "λ", or equivalently the maximum number of linearly independent eigenvectors associated with "λ", is referred to as the eigenvalue's geometric multiplicity "γ""A"("λ"). Because "E" is also the nullspace of ("A" − "λI"), the geometric multiplicity of "λ" is the dimension of the nullspace of ("A" − "λI"), also called the "nullity" of ("A" − "λI"), which relates to the dimension and rank of ("A" − "λI") 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 formula_22, consider how the definition of geometric multiplicity implies the existence of formula_23 orthonormal eigenvectors formula_24, such that formula_25. We can therefore find a (unitary) matrix formula_26 whose first formula_23 columns are these eigenvectors, and whose remaining columns can be any orthonormal set of formula_28 vectors orthogonal to these eigenvectors of formula_29. Then formula_26 has full rank and is therefore invertible, and formula_31 with formula_32 a matrix whose top left block is the diagonal matrix formula_33. This implies that formula_34. In other words, formula_35 is similar to formula_36, which implies that formula_37. But from the definition of formula_32 we know that formula_39 contains a factor formula_40, which means that the algebraic multiplicity of formula_1 must satisfy formula_42.
Suppose formula_29 has formula_44 distinct eigenvalues formula_45, where the geometric multiplicity of formula_46 is formula_47. The total geometric multiplicity of formula_29,
is the dimension of the sum of all the eigenspaces of formula_29's eigenvalues, or equivalently the maximum number of linearly independent eigenvectors of formula_29. If formula_52, then
Additional properties of eigenvalues.
Let formula_29 be an arbitrary formula_61 matrix of complex numbers with eigenvalues formula_62. Each eigenvalue appears formula_63 times in this list, where formula_63 is the eigenvalue's algebraic multiplicity. The following are properties of this matrix and its eigenvalues:
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 formula_61 matrix formula_29 in the defining equation, Equation (),
The eigenvalue and eigenvector problem can also be defined for "row" vectors that "left" multiply matrix formula_29. In this formulation, the defining equation is
where formula_101 is a scalar and formula_102 is a formula_103 matrix. Any row vector formula_102 satisfying this equation is called a left eigenvector of formula_29 and formula_101 is its associated eigenvalue. Taking the transpose of this equation,
Comparing this equation to Equation (), it follows immediately that a left eigenvector of formula_29 is the same as the transpose of a right eigenvector of formula_109, with the same eigenvalue. Furthermore, since the characteristic polynomial of formula_109 is the same as the characteristic polynomial of formula_29, the eigenvalues of the left eigenvectors of formula_29 are the same as the eigenvalues of the right eigenvectors of formula_109.
Diagonalization and the eigendecomposition.
Suppose the eigenvectors of "A" form a basis, or equivalently "A" has "n" linearly independent eigenvectors v1, v2, …, v"n" with associated eigenvalues "λ"1, "λ"2, …, "λ""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 Λ where each diagonal element Λ"ii" is the eigenvalue associated with the "i"th 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 Λ or "diagonalizable". The matrix "Q" is the change of basis matrix of the similarity transformation. Essentially, the matrices "A" and Λ represent the same linear transformation expressed in two different bases. The eigenvectors are used as the basis when representing the linear transformation as Λ.
Conversely, suppose a matrix "A" is diagonalizable. Let "P" be a non-singular square matrix such that "P"−1"AP" 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.
A matrix that is not diagonalizable is said to be defective. For defective matrices, the notion of eigenvectors generalizes to generalized eigenvectors and the diagonal matrix of eigenvalues generalizes to the Jordan normal form. Over an algebraically closed field, any matrix "A" has a Jordan normal form and therefore admits a basis of generalized eigenvectors and a decomposition into generalized eigenspaces.
Variational characterization.
In the Hermitian case, eigenvalues can be given a variational characterization. The largest eigenvalue of formula_119 is the maximum value of the quadratic form formula_120. A value of formula_121 that realizes that maximum, is an eigenvector.
Matrix examples.
Two-dimensional matrix example.
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 (), and the values of "λ" for which the determinant of the matrix ("A" − "λI") 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 , Equation () becomes,
Any nonzero vector with "v"1 = −"v"2 solves this equation. Therefore,
is an eigenvector of "A" corresponding to "λ" = 1, as is any scalar multiple of this vector.
For , Equation () becomes
Any nonzero vector with "v"1 = "v"2 solves this equation. Therefore,
is an eigenvector of "A" corresponding to "λ" = 3, as is any scalar multiple of this vector.
Thus, the vectors v"λ"=1 and v"λ"=3 are eigenvectors of "A" associated with the eigenvalues and , respectively.
General formula for eigenvalues of a two-dimensional matrix.
The eigenvalues of the real matrix formula_130 are{2} \\
</math>
Simplify radicand:
formula_131
Bring denominator into the square root:
formula_132
Note that the eigenvalues are always real if "b" and "c" have the same sign, since the quantity under the radical must be positive.
Three-dimensional matrix example.
Consider the matrix
The characteristic polynomial of "A" is
The roots of the characteristic polynomial are 2, 1, and 11, which are the only three eigenvalues of "A". These eigenvalues correspond to the eigenvectors and or any nonzero multiple thereof.
Three-dimensional matrix example with complex eigenvalues.
Consider the cyclic permutation matrix
This matrix shifts the coordinates of the vector up by one position and moves the first coordinate to the bottom. Its characteristic polynomial is 1 − "λ"3, whose roots are
where formula_138 is an imaginary unit with 
For the real eigenvalue "λ"1 = 1, any vector with three equal nonzero entries is an eigenvector. For example,
For the complex conjugate pair of imaginary eigenvalues,
Then
and
Therefore, the other two eigenvectors of "A" are complex and are formula_143 and formula_144 with eigenvalues "λ"2 and "λ"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 "μ""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 formula_154 and is therefore 1-dimensional. Similarly, the geometric multiplicity of the eigenvalue 3 is 1 because its eigenspace is spanned by just one vector formula_155. The total geometric multiplicity "γ""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.
Eigenvector-eigenvalue identity.
For a Hermitian matrix, the norm squared of the "j"th component of a normalized eigenvector can be calculated using only the matrix eigenvalues and the eigenvalues of the corresponding minor matrix,
where formula_157 is the submatrix formed by removing the "j"th row and column from the original 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 formula_159 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 "λ" = 0 the eigenfunction "f"("t") is a constant.
The main eigenfunction article gives other examples.
General definition.
The concept of eigenvalues and eigenvectors extends naturally to arbitrary linear transformations on arbitrary vector spaces. Let "V" be any vector space over some field "K" 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 "λ" ∈ "K" such that
This equation is called the eigenvalue equation for "T", and the scalar "λ" 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 "λ"v is the product of the scalar "λ" with v.
Eigenspaces, geometric multiplicity, and the eigenbasis.
Given an eigenvalue "λ", consider the set
which is the union of the zero vector with the set of all eigenvectors associated with "λ". "E" is called the eigenspace or characteristic space of "T" associated with "λ".
By definition of a linear transformation,
for x, y ∈ "V" and "α" ∈ "K". Therefore, if u and v are eigenvectors of "T" associated with eigenvalue "λ", namely u, v ∈ "E", then
So, both u + v and αv are either zero or eigenvectors of "T" associated with "λ", namely u + v, "α"v ∈ "E", and "E" is closed under addition and scalar multiplication. The eigenspace "E" associated with "λ" is therefore a linear subspace of "V".
If that subspace has dimension 1, it is sometimes called an eigenline.
The geometric multiplicity "γ""T"("λ") of an eigenvalue "λ" is the dimension of the eigenspace associated with "λ", i.e., the maximum number of linearly independent eigenvectors associated with that eigenvalue. By the definition of eigenvalues and eigenvectors, "γ""T"("λ") ≥ 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.
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 (). Define an eigenvalue to be any scalar "λ" ∈ "K" such that there exists a nonzero vector v ∈ "V" satisfying Equation (). 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 eigenvector v associated with the eigenvalue "λ" to be any vector that, given "λ", satisfies Equation (). Given the eigenvalue, the zero vector is among the vectors that satisfy Equation (), so the zero vector is included among the eigenvectors by this alternate definition.
Spectral theory.
If "λ" is an eigenvalue of "T", then the operator ("T" − "λI") is not one-to-one, and therefore its inverse ("T" − "λI")−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" − "λI") may not have an inverse even if "λ" is not an eigenvalue.
For this reason, in functional analysis eigenvalues can be generalized to the spectrum of a linear operator "T" as the set of all scalars "λ" for which the operator ("T" − "λI") has no bounded inverse. The spectrum of an operator always contains all its eigenvalues but is not limited to them.
Associative algebras and representation theory.
One can generalize the algebraic object that is acting on the vector space, replacing a single operator acting on a vector space with an algebra representation – an associative algebra acting on a module. The study of such actions is the field of representation theory.
The representation-theoretical concept of weight is an analog of eigenvalues, while "weight vectors" and "weight spaces" are the analogs of eigenvectors and eigenspaces, respectively.
Dynamic equations.
The simplest difference equations have the form
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 formula_168 giving a "k"-dimensional system of the first order in the stacked variable vector formula_169 in terms of its once-lagged value, and taking the characteristic equation of this system's matrix. This equation gives "k" characteristic roots formula_170 for use in the solution equation
A similar procedure is used for solving a differential equation of the form
Calculation.
The calculation of eigenvalues and eigenvectors is a topic where theory, as presented in elementary linear algebra textbooks, is often very far from practice.
Classical method.
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.
Eigenvalues.
The eigenvalues of a matrix formula_29 can be determined by finding the roots of the characteristic polynomial. This is easy for formula_174 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 formula_175 is a sum of formula_176 different products.
Explicit algebraic formulas for the roots of a polynomial exist only if the degree formula_56 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 formula_56 is the characteristic polynomial of some companion matrix of order formula_56.) 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.
Eigenvectors.
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 formula_181, that is
This matrix equation is equivalent to two linear equations
Both equations reduce to the single linear equation formula_185. Therefore, any vector of the form formula_186, for any nonzero real number formula_187, is an eigenvector of formula_29 with eigenvalue formula_189.
The matrix formula_29 above has another eigenvalue formula_191. A similar calculation shows that the corresponding eigenvectors are the nonzero solutions of formula_192, that is, any vector of the form formula_193, for any nonzero real number formula_194.
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 formula_195; this causes it to converge to an eigenvector of the eigenvalue closest to formula_196.
If formula_197 is (a good approximation of) an eigenvector of formula_29, then the corresponding eigenvalue can be computed as
where formula_200 denotes the conjugate transpose of formula_197.
Modern methods.
Efficient, accurate methods to compute eigenvalues and eigenvectors of arbitrary matrices were not known until the QR algorithm was designed in 1961. Combining the Householder transformation with the LU decomposition results in an algorithm with better convergence than the QR algorithm. For large Hermitian sparse matrices, the Lanczos algorithm is one example of an efficient iterative method to compute eigenvalues and eigenvectors, among several other possibilities.
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.
Applications.
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 formula_202, 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, formula_203; 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.
Schrödinger equation.
An example of an eigenvalue equation where the transformation formula_204 is represented in terms of a differential operator is the time-independent Schrödinger equation in quantum mechanics:
where formula_119, the Hamiltonian, is a second-order differential operator and formula_207, the wavefunction, is one of its eigenfunctions corresponding to the eigenvalue formula_208, 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 formula_207 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 formula_207 and formula_119 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 formula_212. In this notation, the Schrödinger equation is:
where formula_212 is an eigenstate of formula_119 and formula_208 represents the eigenvalue. formula_119 is an observable self adjoint operator, the infinite-dimensional analog of Hermitian matrices. As in the matrix case, in the equation above formula_218 is understood to be the vector obtained by application of the transformation formula_119 to formula_212.
Wave transport.
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 formula_221. The eigenvectors of the transmission operator formula_222 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, formula_223, of formula_222 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 formula_225 and formula_226. Furthermore, one of the striking properties of open eigenchannels, beyond the perfect transmittance, is the statistically robust spatial profile of the eigenchannels.
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. Thus, if one wants to underline this aspect, one speaks of nonlinear eigenvalue problems. 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.
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 formula_227 by their eigenvalues formula_228;
formula_229 then is the primary orientation/dip of clast, formula_230 is the secondary and formula_231 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 formula_232, formula_233, and formula_234 are dictated by the nature of the sediment's fabric. If formula_235, the fabric is said to be isotropic. If formula_236, the fabric is said to be planar. If formula_237, the fabric is said to be linear.
Principal component analysis.
The eigendecomposition of a symmetric positive semidefinite (PSD) matrix yields an orthogonal basis of eigenvectors, each of which has a nonnegative eigenvalue. The orthogonal decomposition of a PSD matrix is used in multivariate analysis, where the sample covariance matrices are PSD. This orthogonal decomposition is called principal component analysis (PCA) in statistics. PCA studies linear relations among variables. PCA is performed on the covariance matrix or the correlation matrix (in which each variable is scaled to have its sample variance equal to one). For the covariance or correlation matrix, the eigenvectors correspond to principal components and the eigenvalues to the variance explained by the principal components. Principal component analysis of the correlation matrix provides an orthogonal basis for the space of the observed data: In this basis, the largest eigenvalues correspond to the principal components that are associated with most of the covariability among a number of observed data.
Principal component analysis is used as a means of dimensionality reduction in the study of large data sets, such as those encountered in bioinformatics. In Q methodology, the eigenvalues of the correlation matrix determine the Q-methodologist's judgment of "practical" significance (which differs from the statistical significance of hypothesis testing; cf. criteria for determining the number of factors). More generally, principal component analysis can be used as a method of factor analysis in structural equation modeling.
Vibration analysis.
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
or
that is, acceleration is proportional to position (i.e., we expect formula_240 to be sinusoidal in time).
In formula_56 dimensions, formula_242 becomes a mass matrix and formula_69 a stiffness matrix. Admissible solutions are then a linear combination of solutions to the generalized eigenvalue problem
where formula_245 is the eigenvalue and formula_246 is the (imaginary) angular frequency. The principal vibration modes are different from the principal compliance modes, which are the eigenvectors of formula_69 alone. Furthermore, damped vibration, governed by
leads to a so-called quadratic eigenvalue problem,
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.
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 with a large set of normalized pictures of faces are called eigenfaces; this is an example of principal component analysis. 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.
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.
Tensor of moment of inertia.
In mechanics, the eigenvectors of the moment of inertia tensor define the principal axes of a rigid body. The tensor of moment of inertia is a key quantity required 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.
Graphs.
In spectral graph theory, an eigenvalue of a graph is defined as an eigenvalue of the graph's adjacency matrix formula_29, or (increasingly) of the graph's Laplacian matrix due to its discrete Laplace operator, which is either formula_251 (sometimes called the "combinatorial Laplacian") or formula_252 (sometimes called the "normalized Laplacian"), where formula_32 is a diagonal matrix with formula_254 equal to the degree of vertex formula_255, and in formula_256, the formula_138th diagonal entry is formula_258. The formula_69th principal eigenvector of a graph is defined as either the eigenvector corresponding to the formula_69th largest or formula_69th 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 (formula_262) 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 formula_262 is the average number of people that one typical infectious person will infect. The generation time of an infection is the time, formula_264, 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 formula_264 has passed. formula_262 is then the largest eigenvalue of the next generation matrix.