Eigenvalues and eigenvectors

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In this shear mapping the red arrow changes direction but the blue arrow does not. Therefore the blue arrow is an eigenvector, with eigenvalue 1 as its length is unchanged.

The eigenvectors of a square matrix are the non-zero vectors that, after being multiplied by the matrix, remain parallel to the original vector. For each eigenvector, the corresponding eigenvalue is the factor by which the eigenvector is scaled when multiplied by the matrix. The prefix eigen- is adopted from the German word "eigen" for "own"^[1] in the sense of a characteristic description. The eigenvectors are sometimes also called characteristic vectors. Similarly, the eigenvalues are also known as characteristic values.

The mathematical expression of this idea is as follows: if A is a square matrix, a non-zero vector v is an eigenvector of A if there is a scalar λ (lambda) such that

$A\mathbf{v} = \lambda \mathbf{v} \, .$

The scalar λ (lambda) is said to be the eigenvalue of A corresponding to v. An eigenspace of A is the set of all eigenvectors with the same eigenvalue together with the zero vector. However, the zero vector is not an eigenvector.^[2]

These ideas are often extended to more general situations, where scalars are elements of any field, vectors are elements of any vector space, and linear transformations may or may not be represented by matrix multiplication. For example, instead of real numbers, scalars may be complex numbers; instead of arrows, vectors may be functions or frequencies; instead of matrix multiplication, linear transformations may be operators such as the derivative from calculus. These are only a few of countless examples where eigenvectors and eigenvalues are important.

In such cases, the concept of direction loses its ordinary meaning, and is given an abstract definition. Even so, if that abstract direction is unchanged by a given linear transformation, the prefix "eigen" is used, as in eigenfunction, eigenmode, eigenface, eigenstate, and eigenfrequency.

Eigenvalues and eigenvectors have many applications in both pure and applied mathematics. They are used in matrix factorization, in quantum mechanics, and in many other areas.

1 Definition
2 Eigenvalues and eigenvectors of matrices
3 Calculation
4 History
5 Generalizations
6 Applications
7 See also
8 Notes
9 References
10 External links

[ Definition

[ Prerequisites and motivation

Matrix A acts by stretching the vector x, not changing its direction, so x is an eigenvector of A.

Eigenvectors and eigenvalues depend on the concepts of vectors and linear transformations. In the most elementary case, vectors can be thought of as arrows that have both length (or magnitude) and direction. Once a set of Cartesian coordinates is established, a vector can be described relative to that set of coordinates by a sequence of numbers. A linear transformation can be described by a square matrix. For example, in the standard coordinates of n-dimensional space, a vector can be written

$\mathbf x = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}.$

A matrix can be written

$A = \begin{bmatrix} a_{1,1} & a_{1,2} & \ldots & a_{1,n} \\ a_{2,1} & a_{2,2} & \ldots & a_{2,n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n,1} & a_{n,2} & \ldots & a_{n,n} \\ \end{bmatrix}.$

Here n is a fixed natural number.

Usually, the multiplication of a vector x by a square matrix A changes both the magnitude and the direction of the vector it acts on—but in the special case where it changes only the scale (magnitude) of the vector and leaves the direction unchanged, or switches the vector to the opposite direction, that vector is called an eigenvector of that matrix. (The term "eigenvector" is meaningless except in relation to some particular matrix.) When multiplied by a matrix, each eigenvector of that matrix changes its magnitude by a factor, called the eigenvalue corresponding to that eigenvector.

The vector x is an eigenvector of the matrix A with eigenvalue λ (lambda) if the following equation holds:

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

This equation can be interpreted geometrically as follows: a vector x is an eigenvector if multiplication by A stretches, shrinks, leaves unchanged, flips (points in the opposite direction), flips and stretches, or flips and shrinks x. If the eigenvalue λ > 1, x is stretched by this factor. If λ = 1, the vector x is not affected at all by multiplication by A. If 0 < λ < 1, x is shrunk (or compressed). The case λ = 0 means that x shrinks to a point (represented by the origin), meaning that x is in the kernel of the linear map given by A. If λ < 0 then x flips and points in the opposite direction as well as being scaled by a factor equal to the absolute value of λ.

As a special case, the identity matrix I is the matrix that leaves all vectors unchanged:

$I \mathbf x = 1 \mathbf x = \mathbf x. \,$

Every non-zero vector x is an eigenvector of the identity matrix with eigenvalue 1.

[ Example

For the matrix A

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

the vector

$\mathbf x = \begin{bmatrix} 3 \\ -3 \end{bmatrix}$

is an eigenvector with eigenvalue 1. Indeed,

$A \mathbf x = \begin{bmatrix} 2 & 1\\1 & 2 \end{bmatrix} \begin{bmatrix} 3 \\ -3 \end{bmatrix} = \begin{bmatrix} 2 \cdot 3 + 1 \cdot (-3) \\ 1 \cdot 3 + 2 \cdot (-3) \end{bmatrix} = \begin{bmatrix} 3 \\ -3 \end{bmatrix} = 1 \cdot \begin{bmatrix} 3 \\ -3 \end{bmatrix}.$

On the other hand the vector

$\mathbf x = \begin{bmatrix} 0 \\ 1 \end{bmatrix}$

is not an eigenvector, since

$\begin{bmatrix} 2 & 1\\1 & 2 \end{bmatrix} \begin{bmatrix} 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 2 \cdot 0 + 1 \cdot 1 \\ 1 \cdot 0 + 2 \cdot 1 \end{bmatrix} = \begin{bmatrix} 1 \\ 2 \end{bmatrix}.$

and this vector is not a multiple of the original vector x.

[ Formal definition

In abstract mathematics, a more general definition is given:

Let V be any vector space, let x be a vector in that vector space, and let T be a linear transformation mapping V into V. Then x is an eigenvector of T with eigenvalue λ if the following equation holds:

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

This equation is called the eigenvalue equation. Note that Tx means T of x, the action of the transformation T on x, while λx means the product of the number λ times the vector x.^[3] Most, but not all ^[4] authors also require x to be non-zero. The set of eigenvalues of T is sometimes called the spectrum of T.

[ Eigenvalues and eigenvectors of matrices

[ Characteristic polynomial

Main article: Characteristic polynomial

The eigenvalues of A are precisely the solutions λ to the equation

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

Here det is the determinant of matrix formed by A - λI and I is the n×n identity matrix. This equation is called the characteristic equation (or, less often, the secular equation) of A. For example, if A is the following matrix (a so-called diagonal matrix):

$A = \begin{bmatrix} a_{1,1} & 0 & \cdots & 0 \\ 0 & a_{2,2} & \cdots & 0 \\ \vdots & \vdots & \ddots & 0 \\ 0 & 0 & 0 & a_{n,n} \end{bmatrix},$

then the characteristic equation reads

$\det (A-\lambda I) = \det \left(\begin{bmatrix} a_{1,1} & 0 & \cdots & 0 \\ 0 & a_{2,2} & \cdots & 0 \\ \vdots & \vdots & \ddots & 0 \\ 0 & 0 & 0 & a_{n,n} \end{bmatrix} - \lambda \begin{bmatrix} 1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}\right)$

$\begin{align} & = \det \begin{bmatrix} a_{1,1} - \lambda & 0 & \cdots & 0 \\ 0 & a_{2,2} - \lambda& \cdots & 0 \\ \vdots & \vdots & \ddots & 0 \\ 0 & 0 & 0 & a_{n,n} - \lambda \end{bmatrix} \\ & = (a_{1,1} - \lambda) (a_{2,2} - \lambda) \cdots (a_{n,n}- \lambda) = 0 \end{align}$ .

The solutions to this equation are the eigenvalues λ_i = a_i,i (i = 1, ..., n).

Proving the afore-mentioned relation of eigenvalues and solutions of the characteristic equation requires some linear algebra, specifically the notion of linearly independent vectors: briefly, the eigenvalue equation for a matrix A can be expressed as

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

which can be rearranged to

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

If there exists an inverse

$\displaystyle(A - \lambda I)^{-1},$

then both sides can be left-multiplied by it, to obtain x = 0. Therefore, if λ is such that A − λI is invertible, λ cannot be an eigenvalue. It can be shown that the converse holds, too: if A − λI is not invertible, λ is an eigenvalue. A criterion from linear algebra states that a matrix (here: A − λI) is non-invertible if and only if its determinant is zero, thus leading to the characteristic equation.

The left-hand side of this equation can be seen (using Leibniz' rule for the determinant) to be a polynomial function in λ, whose coefficients depend on the entries of A. This polynomial is called the characteristic polynomial. Its degree is n, that is to say, the highest power of λ occurring in this polynomial is λⁿ. At least for small matrices, the solutions of the characteristic equation (hence, the eigenvalues of A) can be found directly. Moreover, it is important for theoretical purposes, such as the Cayley–Hamilton theorem. It also shows that any n×n matrix has at most n eigenvalues. However, the characteristic equation need not have n distinct solutions. In other words, there may be strictly less than n distinct eigenvalues. This happens for the matrix describing the shear mapping discussed below.

If the matrix has real entries, the coefficients of the characteristic polynomial are all real. However, the roots are not necessarily real; they may include complex numbers with a non-zero imaginary component. For example, a 2×2 matrix describing a 45° rotation will not leave any non-zero vector pointing in the same direction. However, there is at least one complex number λ solving the characteristic equation, even if the entries of the matrix A are complex numbers to begin with. (This existence of such a solution is known as the fundamental theorem of algebra.) For a complex eigenvalue, the corresponding eigenvectors also have complex components.

[ Eigenspace

If x is an eigenvector of the matrix A with eigenvalue λ, then any scalar multiple αx is also an eigenvector of A with the same eigenvalue, since A(αx) = αAx = αλx = λ(αx). More generally, any non-zero linear combination of eigenvectors that share the same eigenvalue λ, will itself be an eigenvector with eigenvalue λ.[5] Together with the zero vector, the eigenvectors of A with the same eigenvalue form a linear subspace of the vector space called an eigenspace, E_λ. In case of dim(E_λ) = 1, it is called an eigenline and λ is called a scaling factor.

Diagonalizable matrices can be decomposed into a direct sum of eigenspaces, as per the eigendecomposition of a matrix. If a matrix is not diagonalizable, then it is called defective, and, while it cannot be decomposed into eigenspaces, it can be decomposed into the more general concept of generalized eigenspaces, as discussed here.

[ Algebraic and geometric multiplicities

Given an n×n matrix A and an eigenvalue λ_i of this matrix, there are two numbers measuring, roughly speaking, the number of eigenvectors belonging to λ_i. They are called multiplicities: the algebraic multiplicity of an eigenvalue is defined as the multiplicity of the corresponding root of the characteristic polynomial. The geometric multiplicity of an eigenvalue is defined as the dimension of the associated eigenspace, i.e. number of linearly independent eigenvectors with that eigenvalue. Both algebraic and geometric multiplicity are integers between (including) 1 and n. The algebraic multiplicity n_i and geometric multiplicity m_i may or may not be equal, but we always have m_i ≤ n_i. The simplest case is of course when m_i = n_i = 1. The total number of linearly independent eigenvectors, N_x, is given by summing the geometric multiplicities

$\sum\limits_{i=1}^{N_{\lambda}}{m_i} =N_{\mathbf{x}}.$

Over a complex vector space, the sum of the algebraic multiplicities will equal the dimension of the vector space, but the sum of the geometric multiplicities may be smaller. In this case, it is possible that there may not be sufficient eigenvectors to span the entire space – more formally, there is no basis of eigenvectors (an eigenbasis). A matrix is diagonalizable by a suitable choice of coordinates if and only if there is an eigenbasis; if a matrix is not diagonalizable, it is said to be defective. For defective matrices, the notion of eigenvector can be generalized to generalized eigenvectors, and over an algebraically closed field a basis of generalized eigenvectors always exists, as follows from Jordan form.

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 eigenvalues (or eigenspaces).^[6] All defective matrices have fewer than n distinct eigenvalues, but not all matrices with fewer than n distinct eigenvalues are defective^[7] – for example, the identity matrix is diagonalizable (and indeed diagonal in any basis), but only has the eigenvalue 1.

Given an ordered choice of linearly independent eigenvectors, especially an eigenbasis, they can be indexed by eigenvalues, i.e. using a double index, with x_i,j being the j ^th eigenvector for the i ^th eigenvalue. The eigenvectors can also be indexed using the simpler notation of a single index x_k, with k = 1, 2, ... , N_x.

[ Worked example

These concepts are explained for the matrix

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

The characteristic equation of this matrix reads

$\det (A - \lambda I) = \det\begin{bmatrix} 2-\lambda & 1\\1 & 2-\lambda \end{bmatrix} = 0 \,.$

Calculating the determinant, this yields the quadratic equation

$\lambda^2 - 4 \lambda + 3 = 0, \,$

whose solutions (also called roots) are $λ = 1$ and $λ = 3$ . The eigenvectors for the eigenvalue $λ = 3$ are determined by using the eigenvalue equation, which in this case reads

$\begin{bmatrix} 2 & 1\\1 & 2 \end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix} = 3 \begin{bmatrix}x\\y\end{bmatrix}.$

The juxtaposition at the left hand side denotes matrix multiplication. Spelling this out, this equation comparing two vectors is tantamount to a system of the following two linear equations:

$2x+y=3x\,$

$x+2y=3y.\,$

Both equations reduce to the single linear equation $x = y$ . That is to say, any vector of the form (x, y) with y = x is an eigenvector to the eigenvalue λ = 3. However, the vector (0, 0) is excluded. A similar calculation shows that the eigenvectors corresponding to the eigenvalue $λ = 1,$ are given by non-zero vectors (x, y) such that y = −x. For example, an eigenvector corresponding to $λ = 1,$ is $\begin{bmatrix}-1\\1\end{bmatrix}$ whereas an eigenvector corresponding to $λ = 3,$ is $\begin{bmatrix}1\\1\end{bmatrix}$ . These vectors, placed as columns in a matrix, may be used to create a diagonalizable matrix.

[ Eigendecomposition

Main article: Eigendecomposition of a matrix

The spectral theorem for matrices can be stated as follows. Let A be a square n × n matrix. Let q₁ ... 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 A. If k = n, then A can be written

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

where Q is the square n × n matrix whose i-th column is the basis eigenvector q_i of A and Λ is the diagonal matrix whose diagonal elements are the corresponding eigenvalues, i.e. Λ_ii = λ_i.

[ Further properties

Let $A$ be an n×n matrix with eigenvalues $λ i$ , $i=1,2,\dots,n$ . Then

Trace of A

$\operatorname{tr}(A) = \sum \lambda_i= \lambda_1+ \lambda_2 +\cdots+ \lambda_n$ .

Determinant of A

$\operatorname{det}(A) = \prod \lambda_i=\lambda_1\lambda_2\cdots\lambda_n$ .

Eigenvalues of $A k$ are $\lambda_1^k,\dots,\lambda_n^k$

These first three results follow by putting the matrix in upper-triangular form, in which case the eigenvalues are on the diagonal and the trace and determinant are respectively the sum and product of the diagonal.

If $A = A H$ , i.e., $A$ is Hermitian, every eigenvalue is real.

Every eigenvalue of a Unitary matrix has absolute value $| λ | = 1$ .

[ Examples in the plane

The following table presents some example transformations in the plane along with their 2×2 matrices, eigenvalues, and eigenvectors.

	horizontal shear	scaling	unequal scaling	counterclockwise rotation by $φ$
illustration
matrix	$\begin{bmatrix}1 & k\\ 0 & 1\end{bmatrix}$	$\begin{bmatrix}k & 0\\0 & k\end{bmatrix}$	$\begin{bmatrix}k_1 & 0\\0 & k_2\end{bmatrix}$	$\begin{bmatrix} \cos \varphi & -\sin \varphi \\ \sin \varphi & \cos \varphi \end{bmatrix}$
characteristic equation	λ² − 2λ+1 = (1 − λ)² = 0	λ² − 2λk + k² = (λ − k)² = 0	(λ − k₁)(λ − k₂) = 0	λ² − 2λ cos φ + 1 = 0
eigenvalues λ_i	λ₁=1	λ₁=k	λ₁ = k₁, λ₂ = k₂	λ_1,2 = cos φ ± i sin φ = e ^{± iφ}
algebraic and geometric multiplicities	n₁ = 2, m₁ = 1	n₁ = 2, m₁ = 2	n₁ = m₁ = 1, n₂ = m₂ = 1	n₁ = m₁ = 1, n₂ = m₂ = 1
eigenvectors	$\mathbf u_1 = (1, 0)$	$\mathbf u_1 = (1, 0), \mathbf u_2 = (0,1)$	$\mathbf u_1 = (1, 0), \mathbf u_2 = (0,1)$	$\mathbf u_1 = \begin{bmatrix}1\\-i\end{bmatrix}, \mathbf u_2 = \begin{bmatrix}1\\i\end{bmatrix}.$

[ Shear

Shear in the plane is a transformation where all points along a given line remain fixed while other points are shifted parallel to that line by a distance proportional to their perpendicular distance from the line.[8] In the horizontal shear depicted above, a point P of the plane moves parallel to the x-axis to the place P' so that its coordinate y does not change while the x coordinate increments to become x' = x + k y, where k is called the shear factor. The shear angle φ is determined by k = cot φ.

Repeatedly applying the shear transformation changes the direction of any vector in the plane closer and closer to the direction of the eigenvector.

[ Uniform scaling and reflection

Multiplying every vector with a constant real number k is represented by the diagonal matrix whose entries on the diagonal are all equal to k. Mechanically, this corresponds to stretching a rubber sheet equally in all directions such as a small area of the surface of an inflating balloon. All vectors originating at origin (i.e., the fixed point on the balloon surface) are stretched equally with the same scaling factor k while preserving its original direction. Thus, every non-zero vector is an eigenvector with eigenvalue k. Whether the transformation is stretching (elongation, extension, inflation), or shrinking (compression, deflation) depends on the scaling factor: if k > 1, it is stretching; if 0 < k < 1, it is shrinking. Negative values of k correspond to a reversal of direction, followed by a stretch or a shrink, depending on the absolute value of k.

[ Unequal scaling

For a slightly more complicated example, consider a sheet that is stretched unequally in two perpendicular directions along the coordinate axes, or, similarly, stretched in one direction, and shrunk in the other direction. In this case, there are two different scaling factors: k₁ for the scaling in direction x, and k₂ for the scaling in direction y. If a given eigenvalue is greater than 1, the vectors are stretched in the direction of the corresponding eigenvector; if less than 1, they are shrunken in that direction. Negative eigenvalues correspond to reflections followed by a stretch or shrink. In general, matrices that are diagonalizable over the real numbers represent scalings and reflections: the eigenvalues represent the scaling factors (and appear as the diagonal terms), and the eigenvectors are the directions of the scalings.

The figure shows the case where $k 1 > 1$ and $1 > k 2 > 0$ . The rubber sheet is stretched along the x axis and simultaneously shrunk along the y axis. After repeatedly applying this transformation of stretching/shrinking many times, almost any vector on the surface of the rubber sheet will be oriented closer and closer to the direction of the x axis (the direction of stretching). The exceptions are vectors along the y-axis, which will gradually shrink away to nothing.

[ Rotation

For more details on this topic, see Rotation matrix.

A rotation in a plane is a transformation that describes motion of a vector, plane, coordinates, etc., around a fixed point. Clearly, for rotations other than through 0° and 180°, every vector in the real plane will have its direction changed, and thus there cannot be any eigenvectors. But this is not necessarily true if we consider the same matrix over a complex vector space. The characteristic equation is a quadratic equation with discriminant D = 4 (cos² φ − 1) = − 4 sin² φ, which is a negative number whenever φ is not equal to a multiple of 180°. A rotation of 0°, 360°, … is just the identity transformation (a uniform scaling by +1), while a rotation of 180°, 540°, …, is a reflection (uniform scaling by -1). Otherwise, as expected, there are no real eigenvalues or eigenvectors for rotation in the plane. Instead, the eigenvalues are complex numbers in general. Although not diagonalizable over the reals, the rotation matrix is diagonalizable over the complex numbers, and again the eigenvalues appear on the diagonal. Thus rotation matrices acting on complex spaces can be thought of as scaling matrices, with complex scaling factors.

[ Calculation

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). There are exact solutions for dimensions below 5, but for dimensions greater than or equal to 5 there are generally no exact solutions and one has to resort to numerical methods to find them approximately. (In fact, since the roots of any polynomial can be expressed as eigenvalues of a companion matrix, the Abel–Ruffini theorem implies that there is no general algebraic solution for eigenvalues of 5×5 or larger matrices: any general eigenvalue algorithm is necessarily approximate, although in practice one can obtain any desired accuracy.^[9]) Worse, any computational procedure that starts by computing the coefficients of the characteristic polynomial can be very inaccurate in the presence of round-off error, because the roots of a polynomial are an extremely sensitive function of the coefficients (see Wilkinson's polynomial).^[9] Efficient, accurate methods to compute eigenvalues and eigenvectors of arbitrary matrices were not known until the advent of the QR algorithm in 1961. ^[9] Besides, combining Householder transformation with LU decomposition can get better convergence than QR algorithm.^[10] 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.^[9]

[ 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 studied the rotational motion of a rigid body and discovered the importance of the principal axes. Lagrange realized that the principal axes are the eigenvectors of the inertia matrix.^[11] In the early 19th century, Cauchy saw how their work could be used to classify the quadric surfaces, and generalized it to arbitrary dimensions.^[12] Cauchy also coined the term racine caractéristique (characteristic root) for what is now called eigenvalue; his term survives in characteristic equation.^[13]

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.^[14] 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.^[12] This was extended by Hermite in 1855 to what are now called Hermitian matrices.^[13] Around the same time, Brioschi proved that the eigenvalues of orthogonal matrices lie on the unit circle,^[12] and Clebsch found the corresponding result for skew-symmetric matrices.^[13] Finally, Weierstrass clarified an important aspect in the stability theory started by Laplace by realizing that defective matrices can cause instability.^[12]

In the meantime, Liouville studied eigenvalue problems similar to those of Sturm; the discipline that grew out of their work is now called Sturm–Liouville theory.^[15] 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.^[16]

At the start of the 20th century, Hilbert studied the eigenvalues of integral operators by viewing the operators as infinite matrices.^[17] 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. For some time, the standard term in English was "proper value", but the more distinctive term "eigenvalue" is standard today.^[18]

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 John G.F. Francis^[19] and Vera Kublanovskaya^[20] in 1961.^[21]

[ Generalizations

[ Left and right eigenvectors

The word eigenvector formally refers to the right eigenvector $x R$ . It is defined by the above eigenvalue equation

$A x_R = \lambda_R x_R,\$

and is the most commonly used eigenvector. However, the left eigenvector $x L$ exists as well, and is defined by

$x_L A = \lambda_L x_L. \$

[ Infinite-dimensional spaces and spectral theory

For more details on this topic, see Spectral theorem.

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 (T − λI)⁻¹ is not defined; that is, such that T − λI 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 that have no eigenvectors at all. This can be seen in thee following example. The CC-BY-SA.

Eigenvalue