Eigenvalue, eigenvector and eigenspace
In mathematics, given a linear transformation, an eigenvector of that linear transformation is a nonzero vector which, when applied to that transformation, changes in length, but not direction.
For each eigenvector of a linear transformation, there is a corresponding scalar value called an eigenvalue for that vector, which determines the amount the eigenvector is scaled under the linear transformation. For example, an eigenvalue of +2 means that the eigenvector is doubled in length and points in the same direction. An eigenvalue of +1 means that the eigenvector is unchanged, while an eigenvalue of −1 means that the eigenvector is reversed in direction. An eigenspace of a given transformation for a particular eigenvalue is the set (linear span) of the eigenvectors associated to this eigenvalue, together with the zero vector (which has no direction).
In linear algebra, every linear transformation between finite-dimensional vector spaces can be given by a matrix, which is a rectangular array of numbers arranged in rows and columns. Standard methods for finding eigenvalues, eigenvectors, and eigenspaces of a given matrix are discussed below.
These concepts play a major role in several branches of both pure and applied mathematics — appearing prominently in linear algebra, functional analysis, and to a lesser extent in nonlinear mathematics.
Many kinds of mathematical objects can be treated as vectors: functions, harmonic modes, quantum states, and frequencies, for example. In these cases, the concept of direction loses its ordinary meaning, and is given an abstract definition. Even so, if this abstract direction is unchanged by a given linear transformation, the prefix "eigen" is used, as in eigenfunction, eigenmode, eigenstate, and eigenfrequency.
Contents
History
Eigenvalues are often introduced in the context of linear algebra or matrix theory. Historically, however, they arose in the study of quadratic forms and differential equations.
Euler had also studied the rotational motion of a rigid body and discovered the importance of the principal axes. As Lagrange realized, the principal axes are the eigenvectors of the inertia matrix.^{[1]} In the early 19th century, Cauchy saw how their work could be used to classify the quadric surfaces, and generalized it to arbitrary dimensions.^{[2]} Cauchy also coined the term racine caractéristique (characteristic root) for what is now called eigenvalue; his term survives in characteristic equation.^{[3]}
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.^{[4]} Sturm developed Fourier's ideas further and he brought them to the attention of Cauchy, who combined them with his own ideas and arrived at the fact that symmetric matrices have real eigenvalues.^{[2]} This was extended by Hermite in 1855 to what are now called Hermitian matrices.^{[3]} Around the same time, Brioschi proved that the eigenvalues of orthogonal matrices lie on the unit circle,^{[2]} and Clebsch found the corresponding result for skew-symmetric matrices.^{[3]} Finally, Weierstrass clarified an important aspect in the stability theory started by Laplace by realizing that defective matrices can cause instability.^{[2]}
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.^{[5]} 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.^{[6]}
At the start of the 20th century, Hilbert studied the eigenvalues of integral operators by viewing the operators as infinite matrices.^{[7]} He was the first to use the German word eigen to denote eigenvalues and eigenvectors in 1904, though he may have been following a related usage by Helmholtz. "Eigen" can be translated as "own", "peculiar to", "characteristic", or "individual" — emphasizing how important eigenvalues are to defining the unique nature of a specific transformation. For some time, the standard term in English was "proper value", but the more distinctive term "eigenvalue" is standard today.^{[8]}
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^{[9]}^{[10]} and Vera Kublanovskaya^{[11]} in 1961.^{[12]}
Definitions
- See also: Eigenplane
Linear transformations of a vector space, such as rotation, reflection, stretching, compression, shear or any combination of these, may be visualized by the effect they produce on vectors. In other words, they are vector functions. More formally, in a vector space L, a vector function A is defined if for each vector x of L there corresponds a unique vector y = A(x) of L. For the sake of brevity, the parentheses around the vector on which the transformation is acting are often omitted. A vector function A is linear if it has the following two properties:
- Additivity: A(x + y) = Ax + Ay
- Homogeneity: A(αx) = αAx
where x and y are any two vectors of the vector space L and α is any scalar.^{[13]} Such a function is variously called a linear transformation, linear operator, or linear endomorphism on the space L.
Given a linear transformation A, a non-zero vector x is defined to be an eigenvector of the transformation if it satisfies the eigenvalue equation Ax = λx for some scalar λ. In this situation, the scalar λ is called an eigenvalue of A corresponding to the eigenvector x.^{[14]} |
The key equation in this definition is the eigenvalue equation, Ax = λx. Most vectors x will not satisfy such an equation. A typical vector x changes direction when acted on by A, so that Ax is not a multiple of x. This means that only certain special vectors x are eigenvectors, and only certain special numbers λ are eigenvalues. Of course, if A is a multiple of the identity matrix, then no vector changes direction, and all non-zero vectors are eigenvectors.
The requirement that the eigenvector be non-zero is imposed because the equation A0 = λ0 holds for every A and every λ. Since the equation is always trivially true, it is not an interesting case. In contrast, an eigenvalue can be zero in a nontrivial way. Each eigenvector is associated with a specific eigenvalue. One eigenvalue can be associated with several or even with infinite number of eigenvectors.
Geometrically (Fig. 2), the eigenvalue equation means that under the transformation A eigenvectors experience only changes in magnitude and sign — the direction of Ax is the same as that of x. The eigenvalue λ is simply the amount of "stretch" or "shrink" to which a vector is subjected when transformed by A. If λ = 1, the vector remains unchanged (unaffected by the transformation). A transformation I under which a vector x remains unchanged, Ix = x, is defined as identity transformation. If λ = –1, the vector flips to the opposite direction (rotates to 180°); this is defined as reflection.
If x is an eigenvector of the linear transformation A with eigenvalue λ, then any scalar multiple αx is also an eigenvector of A with the same eigenvalue. Similarly if more than one eigenvector share the same eigenvalue λ, any linear combination of these eigenvectors will itself be an eigenvector with eigenvalue λ. ^{[15]}. Together with the zero vector, the eigenvectors of A with the same eigenvalue form a linear subspace of the vector space called an eigenspace.
The eigenvectors corresponding to different eigenvalues are linearly independent^{[16]} meaning, in particular, that in an n-dimensional space the linear transformation A cannot have more than n eigenvectors with different eigenvalues.^{[17]}
If a basis is defined in vector space, all vectors can be expressed in terms of components. For finite dimensional vector spaces with dimension n, linear transformations can be represented with n × n square matrices. Conversely, every such square matrix corresponds a linear transformation for a given basis. Thus, in a the two-dimensional vector space R^{2} fitted with standard basis, the eigenvector equation for a linear transformation A can be written in the following matrix representation:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \lambda \begin{bmatrix} x \\ y \end{bmatrix},
where the juxtaposition of matrices means matrix multiplication.
Left and right eigenvectors
The word eigenvector formally refers to the right eigenvector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): x_R . It is defined by the above eigenvalue equation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): A x_R = \lambda_R x_R , and is the most commonly used eigenvector. However, the left eigenvector Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): x_L exists as well, and is defined by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): x_L A = \lambda_L x_L .
Characteristic equation
When a transformation is represented by a square matrix A, the eigenvalue equation can be expressed as
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): A \mathbf{x} - \lambda I \mathbf{x} = \mathbf{0}.
It is known from linear algebra that this equation has a non-zero solution for x if and only if the determinantal equation
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \det(A - \lambda I) = 0
holds. This equation is called the characteristic equation (less often, secular equation) of A, and the left-hand side is called the characteristic polynomial. When expanded, this gives a polynomial equation for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \lambda . The eigenvector x or its components are not present in the characteristic equation.
Example
The matrix
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \begin{bmatrix} 2 & 1\\1 & 2 \end{bmatrix}
defines a linear transformation of the real plane. The eigenvalues of this transformation are given by the characteristic equation
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \det\begin{bmatrix} 2-\lambda & 1\\1 & 2-\lambda \end{bmatrix} = (2-\lambda)^2 - 1 = 0.
The roots of this equation (i.e. the values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \lambda for which the equation holds) are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \lambda=1 and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \lambda=3 . Having found the eigenvalues, it is possible to find the eigenvectors. Considering first the eigenvalue Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \lambda=3 , we have
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \begin{bmatrix} 2 & 1\\1 & 2 \end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix} = 3 \begin{bmatrix}x\\y\end{bmatrix}.
Both rows of this matrix equation reduce to the single linear equation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): x=y . To find an eigenvector, we are free to choose any value for x, so by picking x=1 and setting y=x, we find the eigenvector to be
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \begin{bmatrix}1\\1\end{bmatrix}.
We can check this is an eigenvector by checking that :Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \begin{bmatrix}2&1\\1&2\end{bmatrix}\begin{bmatrix}1\\1\end{bmatrix} = \begin{bmatrix}3\\3\end{bmatrix} . For the eigenvalue Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \lambda=1 , a similar process leads to the equation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): x=-y , and hence the eigenvector is given by
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \begin{bmatrix}1\\-1\end{bmatrix}.
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 higher dimensions there are no exact solutions and one has to resort to numerical methods to find them approximately. For large symmetric sparse matrices, Lanczos algorithm is used to compute eigenvalues and eigenvectors.
Existence and multiplicity of eigenvalues
For transformations on real vector spaces, the coefficients of the characteristic polynomial are all real. However, the roots are not necessarily real; they may well be complex numbers, or a mixture of real and complex numbers. For example, a matrix representing a planar rotation of 45 degrees will not leave any non-zero vector pointing in the same direction. Over a complex vector space, the fundamental theorem of algebra guarantees that the characteristic polynomial has at least one root, and thus the linear transformation has at least one eigenvalue.
As well as distinct roots, the characteristic equation may also have repeated roots. However, having repeated roots does not imply there are multiple distinct (i.e. linearly independent) eigenvectors with that eigenvalue. 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.
Over a complex 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 a sense, then it is possible that there may not be sufficient eigenvectors to span the entire space. This is intimately related to the question of whether a given matrix may be diagonalized by a suitable choice of coordinates.
Shear
Shear in the plane is a transformation in which 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.^{[18]} Shearing a plane figure does not change its area. Shear can be horizontal − along the X axis, or vertical − along the Y axis. In horizontal shear (see figure), 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 matrix of a horizontal shear transformation is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \begin{bmatrix}1 & k\\ 0 & 1\end{bmatrix} . The characteristic equation is λ^{2} − 2 λ + 1 = (1 − λ)^{2} = 0 which has a single, repeated root λ = 1. Therefore, the eigenvalue λ = 1 has algebraic multiplicity 2. The eigenvector(s) are found as solutions of
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \begin{bmatrix}1 - 1 & -k\\ 0 & 1 - 1 \end{bmatrix}\begin{bmatrix}x\\ y\end{bmatrix} = \begin{bmatrix}0 & -k\\ 0 & 0 \end{bmatrix}\begin{bmatrix}x\\ y\end{bmatrix} = -ky = 0.
The last equation is equivalent to y = 0, which is a straight line along the x axis. This line represents the one-dimensional eigenspace. In the case of shear the algebraic multiplicity of the eigenvalue (2) is greater than its geometric multiplicity (1, the dimension of the eigenspace). The eigenvector is a vector along the x axis. The case of vertical shear with transformation matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \begin{bmatrix}1 & 0\\ k & 1\end{bmatrix} is dealt with in a similar way; the eigenvector in vertical shear is along the y axis. Applying repeatedly 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
As a one-dimensional vector space, consider a rubber string tied to unmoving support in one end, such as that on a child's sling. Pulling the string away from the point of attachment stretches it and elongates it by some scaling factor λ which is a real number. Each vector on the string is stretched equally, with the same scaling factor λ, and although elongated, it preserves its original direction. For a two-dimensional vector space, consider a rubber sheet stretched equally in all directions such as a small area of the surface of an inflating balloon (Fig. 3). All vectors originating at the fixed point on the balloon surface (the origin) are stretched equally with the same scaling factor λ. This transformation in two-dimensions is described by the 2×2 square matrix:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): A \mathbf{x} = \begin{bmatrix}\lambda & 0\\0 & \lambda\end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix}\lambda \cdot x + 0 \cdot y \\0 \cdot x + \lambda \cdot y\end{bmatrix} = \lambda \begin{bmatrix} x \\ y \end{bmatrix} = \lambda \mathbf{x}.
Expressed in words, the transformation is equivalent to multiplying the length of any vector by λ while preserving its original direction. Since the vector taken was arbitrary, every non-zero vector in the vector space is an eigenvector. Whether the transformation is stretching (elongation, extension, inflation), or shrinking (compression, deflation) depends on the scaling factor: if λ > 1, it is stretching; if λ < 1, it is shrinking. Negative values of λ correspond to a reversal of direction, followed by a stretch or a shrink, depending on the absolute value of λ.
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: k_{1} for the scaling in direction x, and k_{2} for the scaling in direction y. The transformation matrix is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \begin{bmatrix}k_1 & 0\\0 & k_2\end{bmatrix} , and the characteristic equation is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): (k_1-\lambda)(k_2-\lambda) = 0 . The eigenvalues, obtained as roots of this equation are λ_{1} = k_{1}, and λ_{2} = k_{2} which means, as expected, that the two eigenvalues are the scaling factors in the two directions. Plugging k_{1} back in the eigenvalue equation gives one of the eigenvectors:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \begin{bmatrix}0 & 0\\0 & k_2 - k_1\end{bmatrix} \begin{bmatrix} x \\ y\end{bmatrix} = \begin{bmatrix}0\\0\end{bmatrix} or, more simply, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): y=0 .
Thus, the eigenspace is the x-axis. Similarly, substituting Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \lambda=k_2 shows that the corresponding eigenspace is the y-axis. In this case, both eigenvalues have algebraic and geometric multiplicities equal to 1. 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): k_1>1 and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): 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.
A counterclockwise rotation in the horizontal plane about the origin at an angle φ is represented by the matrix
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \mathbf{R} = \begin{bmatrix} \cos \varphi & -\sin \varphi \\ \sin \varphi & \cos \varphi \end{bmatrix}.
The characteristic equation of R is λ^{2} − 2λ cos φ + 1 = 0. This quadratic equation has a discriminant D = 4 (cos^{2} φ − 1) = − 4 sin^{2} φ which is a negative number whenever φ is not equal 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.
Rotation matrices on complex vector spaces
The characteristic equation has two complex roots λ_{1} and λ_{2}. If we choose to think of the rotation matrix as a linear operator on the complex two dimensional, we can consider these complex eigenvalues. The roots are complex conjugates of each other: λ_{1,2} = cos φ ± i sin φ = e^{ ± iφ}, each with an algebraic multiplicity equal to 1, where i is the imaginary unit.
The first eigenvector is found by substituting the first eigenvalue, λ_{1}, back in the eigenvalue equation:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \begin{bmatrix} \cos \varphi - \lambda_1 & -\sin \varphi \\ \sin \varphi & \cos \varphi - \lambda_1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} - i \sin \varphi & -\sin \varphi \\ \sin \varphi & - i \sin \varphi \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}.
The last equation is equivalent to the single equation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): x=iy , and again we are free to set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): x=1 to give the eigenvector
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \begin{bmatrix}1\\-i\end{bmatrix}.
Similarly, substituting in the second eigenvalue gives the single equation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): x=-iy and so the eigenvector is given by
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \begin{bmatrix}1\\i\end{bmatrix}.
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.
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 − λ)^{−1} is not defined; that is, such that T − λ has no bounded inverse.
Clearly if λ is an eigenvalue of T, λ is in the spectrum of T. In general, the converse is not true. There are operators on Hilbert or Banach spaces which have no eigenvectors at all. This can be seen in the following example. The bilateral shift on the Hilbert space ℓ ^{2}(Z) (that is, the space of all sequences of scalars … a_{−1}, a_{0}, a_{1}, a_{2}, … such that
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \cdots + |a_{-1}|^2 + |a_0|^2 + |a_1|^2 + |a_2|^2 + \cdots
converges) has no eigenvalue but does have spectral values.
In infinite-dimensional spaces, the spectrum of a bounded operator is always nonempty. This is also true for an unbounded self adjoint operator. Via its spectral measures, the spectrum of any self adjoint operator, bounded or otherwise, can be decomposed into absolutely continuous, pure point, and singular parts. (See Decomposition of spectrum.)
The hydrogen atom is an example where both types of spectra appear. The eigenfunctions of the hydrogen atom Hamiltonian are called eigenstates and are grouped into two categories. The bound states of the hydrogen atom correspond to the discrete part of the spectrum (they have a discrete set of eigenvalues which can be computed by Rydberg formula) while the ionization processes are described by the continuous part (the energy of the collision/ionization is not quantified).
Eigenfunctions
A common example of such maps on infinite dimensional spaces are the action of differential operators on function spaces. As an example, on the space of infinitely differentiable functions, the process of differentiation defines a linear operator since
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \displaystyle\frac{d}{dt}(af+bg) = a \frac{df}{dt} + b \frac{dg}{dt},
where f(t) and g(t) are differentiable functions, and a and b are constants).
The eigenvalue equation for linear differential operators is then a set of one or more differential equations. The eigenvectors are commonly called eigenfunctions. The most simple case is the eigenvalue equation for differentiation of a real valued function by a single real variable. In this case, the eigenvalue equation becomes the linear differential equation
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \displaystyle\frac{d}{dx} f(x) = \lambda f(x).
Here λ is the eigenvalue associated with the function, f(x). This eigenvalue equation has a solution for all values of λ. If λ is zero, the solution is
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): f(x) = A,\,
where A is any constant; if λ is non-zero, the solution is the exponential function
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): f(x) = Ae^{\lambda x}.
If we expand our horizons to complex valued functions, the value of λ can be any complex number. The spectrum of d/dt is therefore the whole complex plane. This is an example of a continuous spectrum.
Waves on a string
The displacement, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): h(x,t) , of a stressed rope fixed at both ends, like the vibrating strings of a string instrument, satisfies the wave equation
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \frac{\partial^2 h}{\partial t^2} = c^2\frac{\partial^2 h}{\partial x^2},
which is a linear partial differential equation, where c is the constant wave speed. The normal method of solving such an equation is separation of variables. If we assume that h can be written as the product of the form X(x)T(t), we can form a pair of ordinary differential equations:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): X''=-\frac{\omega^2}{c^2}X and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): T''=-\omega^2 T.
Each of these is an eigenvalue equation (the unfamiliar form of the eigenvalue is chosen merely for convenience). For any values of the eigenvalues, the eigenfunctions are given by
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): X = \sin(\frac{\omega x}{c} + \phi) and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): T = \sin(\omega t + \psi).
If we impose boundary conditions -- that the ends of the string are fixed with X(x)=0 at x=0 and x=L, for example -- we can constrain the eigenvalues. For those boundary conditions, we find
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \sin(\phi) = 0 , and so the phase angle Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \phi=0
and
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \sin\left(\frac{\omega L}{c}\right) = 0.
Thus, the constant Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \omega is constrained to take one of the values Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \omega_n = \frac{nc\pi}{L} , where n is any integer. Thus the clamped string supports a family of standing waves of the form
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): h(x,t) = \sin(n\pi x/L)\sin(\omega_n t).
From the point of view of our musical instrument, the frequency Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \omega_n is the frequency of the nth harmonic overtone.
Eigendecomposition
The spectral theorem for matrices can be stated as follows. Let A be a square n × n matrix. Let q_{1} ... 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
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \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}.
Applications
Schrödinger equation
An example of an eigenvalue equation where the transformation T is represented in terms of a differential operator is the time-independent Schrödinger equation in quantum mechanics:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): H\psi_E = E\psi_E \,
where H, the Hamiltonian, is a second-order differential operator and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \psi_E , the wavefunction, is one of its eigenfunctions corresponding to the eigenvalue E, interpreted as its energy.
However, in the case where one is interested only in the bound state solutions of the Schrödinger equation, one looks for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \psi_E within the space of square integrable functions. Since this space is a Hilbert space with a well-defined scalar product, one can introduce a basis set in which Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): \psi_E and H can be represented as a one-dimensional array and a matrix respectively. This allows one to represent the Schrödinger equation in a matrix form. (Fig. 8 presents the lowest eigenfunctions of the Hydrogen atom Hamiltonian.)
The Dirac notation is often used in this context. A vector, which represents a state of the system, in the Hilbert space of square integrable functions is represented by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): |\Psi_E\rangle . In this notation, the Schrödinger equation is:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): H|\Psi_E\rangle = E|\Psi_E\rangle
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): |\Psi_E\rangle is an eigenstate of H. It is a self adjoint operator, the infinite dimensional analog of Hermitian matrices (see Observable). As in the matrix case, in the equation above Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): H|\Psi_E\rangle is understood to be the vector obtained by application of the transformation H to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): |\Psi_E\rangle .
Molecular orbitals
In quantum mechanics, and in particular in atomic and molecular physics, within the Hartree-Fock theory, the atomic and molecular orbitals can be defined by the eigenvectors of the Fock operator. The corresponding eigenvalues are interpreted as ionization potentials via Koopmans' theorem. In this case, the term eigenvector is used in a somewhat more general meaning, since the Fock operator is explicitly dependent on the orbitals and their eigenvalues. If one wants to underline this aspect one speaks of nonlinear eigenvalue problem. 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 ^{[19]}, ^{[20]}, or as a Stereonet on a Wulff Net ^{[21]}. The output for the orientation tensor is in the three orthogonal (perpendicular) axes of space. Eigenvectors output from programs such as Stereo32 ^{[22]} are in the order E_{1} ≥ E_{2} ≥ E_{3}, with E_{1} being the primary orientation of clast orientation/dip, E_{2} being the secondary and E_{3} being the tertiary, in terms of strength. The clast orientation is defined as the eigenvector, on a compass rose of 360°. Dip is measured as the eigenvalue, the modulus of the tensor: this is valued from 0° (no dip) to 90° (vertical). The relative values of E_{1}, E_{2}, and E_{3} are dictated by the nature of the sediment's fabric. If E_{1} = E_{2} = E_{3}, the fabric is said to be isotropic. If E_{1} = E_{2} > E_{3} the fabric is planar. If E_{1} > E_{2} > E_{3} the fabric is linear. See 'A Practical Guide to the Study of Glacial Sediments' by Benn & Evans, 2004 ^{[23]}.
Factor analysis
In factor analysis, the eigenvectors of a covariance matrix or correlation matrix correspond to factors, and eigenvalues to the variance explained by these factors. Factor analysis is a statistical technique used in the social sciences and in marketing, product management, operations research, and other applied sciences that deal with large quantities of data. The objective is to explain most of the covariability among a number of observable random variables in terms of a smaller number of unobservable latent variables called factors. The observable random variables are modeled as linear combinations of the factors, plus unique variance terms. Eigenvalues are used in analysis used by Q-methodology software; factors with eigenvalues greater than 1.00 are considered significant, explaining an important amount of the variability in the data, while eigenvalues less than 1.00 are considered too weak, not explaining a significant portion of the data variability.
Vibration analysis
Eigenvalue problems occur naturally in the vibration analysis of mechanical structures with many degrees of freedom. The eigenvalues are used to determine the natural frequencies of vibration, and the eigenvectors determine the shapes of these vibrational modes. 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.
The mode shapes of a cantilevered I-beam | ||
---|---|---|
Eigenfaces
In image processing, processed images of faces can be seen as vectors whose components are the brightnesses of each pixel.^{[24]} The dimension of this vector space is the number of pixels. The eigenvectors of the covariance matrix associated to a large set of normalized pictures of faces are called eigenfaces; this is an example of principal components 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. More on determining sign language letters using eigen systems can be found here: http://www.geigel.com/signlanguage/index.php
Similar to this concept, eigenvoices concept is also developed which represents the general direction of variability in human pronunciations of a particular utterance, such as a word in a language. Based on a linear combination of such eigenvoices, a new voice pronunciation of the word can be constructed. These concepts have been found useful in automatic speech recognition systems, for speaker adaptation.
Tensor of inertia
In mechanics, the eigenvectors of the inertia tensor define the principal axes of a rigid body. The tensor of inertia is a key quantity required in order to determine the rotation of a rigid body around its center of mass.
Stress tensor
In solid mechanics, the stress tensor is symmetric and so can be decomposed into a diagonal tensor with the eigenvalues on the diagonal and eigenvectors as a basis. Because it is diagonal, in this orientation, the stress tensor has no shear components; the components it does have are the principal components.
Eigenvalues of a graph
In spectral graph theory, an eigenvalue of a graph is defined as an eigenvalue of the graph's adjacency matrix A, or (increasingly) of the graph's Laplacian matrix, which is either T−A or I−T^{ 1/2}AT^{ −1/2}, where T is a diagonal matrix holding the degree of each vertex, and in T^{ −1/2}, 0 is substituted for 0^{−1/2}. The kth principal eigenvector of a graph is defined as either the eigenvector corresponding to the kth largest eigenvalue of A, or the eigenvector corresponding to the kth smallest eigenvalue of the Laplacian. The first principal eigenvector of the graph is also referred to merely as the principal eigenvector.
The principal eigenvector is used to measure the centrality of its vertices. An example is Google's PageRank algorithm. The principal eigenvector of a modified adjacency matrix of the World Wide Web graph gives the page ranks as its components. This vector corresponds to the stationary distribution of the Markov chain represented by the row-normalized adjacency matrix; however, the adjacency matrix must first be modified to ensure a stationary distribution exists. The second principal eigenvector can be used to partition the graph into clusters, via spectral clustering. Other methods are also available for clustering.
See also
Notes
- ↑ See Hawkins 1975, §2
- ↑ ^{2.0} ^{2.1} ^{2.2} ^{2.3} See Hawkins 1975, §3
- ↑ ^{3.0} ^{3.1} ^{3.2} See Kline 1972, pp. 807-808
- ↑ See Kline 1972, p. 673
- ↑ See Kline 1972, pp. 715-716
- ↑ See Kline 1972, pp. 706-707
- ↑ See Kline 1972, p. 1063
- ↑ See Aldrich 2006
- ↑ J.G.F. Francis, "The QR Transformation, I" (part 1), The Computer Journal, vol. 4, no. 3, pages 265-271 (1961); "The QR Transformation, II" (part 2), The Computer Journal, vol. 4, no. 4, pages 332-345 (1962).
- ↑ John G.F. Francis (1934 - ), devised the “QR transformation” for computing the eigenvalues of matrices. Born in London in 1934, he presently (2007) resides in Hove, England (near Brighton). In 1954 he worked for the National Research Development Corporation (NRDC). In 1955-1956 he attended Cambridge University. He then returned to the NRDC, where he served as assistant to Christopher Strachey. At this time he devised the QR transformation. In 1961 he left the NRDC to work at Ferranti Corporation, Ltd. and then at the University of Sussex. Subsequently, he had positions with various industrial organizations and consultancies. His interests encompassed artificial intelligence, computer languages, and systems engineering. He is currently retired. (See: http://www-sbras.nsc.ru/mathpub/na-net/db/showfile.phtml?v07n34.html#1 .)
- ↑ Vera N. Kublanovskaya, "On some algorithms for the solution of the complete eigenvalue problem" USSR Computational Mathematics and Mathematical Physics, vol. 3, pages 637–657 (1961). Also published in: Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki, vol.1, no. 4, pages 555–570 (1961).
- ↑ See Golub & van Loan 1996, §7.3; Meyer 2000, §7.3
- ↑ See Beezer 2006, Definition LT on p. 507; Strang 2006, p. 117; Kuttler 2007, Definition 5.3.1 on p. 71; Shilov 1977, Section 4.21 on p. 77; Rowland, Todd and Weisstein, Eric W. Linear transformation From MathWorld − A Wolfram Web Resource
- ↑ See Korn & Korn 2000, Section 14.3.5a; Friedberg, Insel & Spence 1989, p. 217
- ↑ For a proof of this lemma, see Shilov 1977, p. 109, and Lemma for the eigenspace
- ↑ For a proof of this lemma, see Roman 2008, Theorem 8.2 on p. 186; Shilov 1977, p. 109; Hefferon 2001, p. 364; Beezer 2006, Theorem EDELI on p. 469; and Lemma for linear independence of eigenvectors
- ↑ See Shilov 1977, p. 109
- ↑ Definition according to Weisstein, Eric W. Shear From MathWorld − A Wolfram Web Resource
- ↑ Graham, D., and Midgley, N., 2000. Earth Surface Processes and Landforms (25) pp 1473-1477
- ↑ Sneed ED, Folk RL. 1958. Pebbles in the lower Colorado River, Texas, a study of particle morphogenesis. Journal of Geology 66(2): 114–150
- ↑ GIS-stereoplot: an interactive stereonet plotting module for ArcView 3.0 geographic information system
- ↑ Stereo32
- ↑ Benn, D., Evans, D., 2004. A Practical Guide to the study of Glacial Sediments. London: Arnold. pp 103-107
- ↑ Xirouhakis, A.; Votsis, G.; Delopoulus, A. (2004), Estimation of 3D motion and structure of human faces (PDF), Online paper in PDF format, National Technical University of Athens
References
- Korn, Granino A.; Korn, Theresa M. (2000), Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review, 1152 p., Dover Publications, 2 Revised edition, ISBN 0-486-41147-8.
- Lipschutz, Seymour (1991), Schaum's outline of theory and problems of linear algebra, Schaum's outline series (2nd ed.), New York, NY: McGraw-Hill Companies, ISBN 0-07-038007-4.
- Friedberg, Stephen H.; Insel, Arnold J.; Spence, Lawrence E. (1989), Linear algebra (2nd ed.), Englewood Cliffs, NJ 07632: Prentice Hall, ISBN 0-13-537102-3.
- Aldrich, John (2006), "Eigenvalue, eigenfunction, eigenvector, and related terms", in Jeff Miller (Editor), Earliest Known Uses of Some of the Words of Mathematics, retrieved 2006-08-22
- Strang, Gilbert (1993), Introduction to linear algebra, Wellesley-Cambridge Press, Wellesley, MA, ISBN 0-961-40885-5.
- Strang, Gilbert (2006), Linear algebra and its applications, Thomson, Brooks/Cole, Belmont, CA, ISBN 0-030-10567-6.
- Bowen, Ray M.; Wang, Chao-Cheng (1980), Linear and multilinear algebra, Plenum Press, New York, NY, ISBN 0-306-37508-7.
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- Fraleigh, John B.; Beauregard, Raymond A. (1995), Linear algebra (3rd ed.), Addison-Wesley Publishing Company, ISBN 0-201-83999-7 (international edition).
- Golub, Gene H.; van Loan, Charles F. (1996), Matrix computations (3rd Edition), Johns Hopkins University Press, Baltimore, MD, ISBN 978-0-8018-5414-9.
- Hawkins, T. (1975), "Cauchy and the spectral theory of matrices", Historia Mathematica, 2: 1–29.
- Horn, Roger A.; Johnson, Charles F. (1985), Matrix analysis, Cambridge University Press, ISBN 0-521-30586-1 (hardback), ISBN 0-521-38632-2 (paperback).
- Kline, Morris (1972), Mathematical thought from ancient to modern times, Oxford University Press, ISBN 0-195-01496-0.
- Meyer, Carl D. (2000), Matrix analysis and applied linear algebra, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, ISBN 978-0-89871-454-8.
- Brown, Maureen (October 2004), Illuminating Patterns of Perception: An Overview of Q Methodology.
- Golub, Gene F.; van der Vorst, Henk A. (2000), "Eigenvalue computation in the 20th century", Journal of Computational and Applied Mathematics, 123: 35–65.
- Akivis, Max A. (1969), Tensor calculus, Russian, Science Publishers, Moscow .
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- Carter, Tamara A.; Tapia, Richard A.; Papaconstantinou, Anne, Linear Algebra: An Introduction to Linear Algebra for Pre-Calculus Students, Rice University, Online Edition, retrieved 2008-02-19.
- Roman, Steven (2008), Advanced linear algebra (3rd ed.), New York, NY: Springer Science + Business Media, LLC, ISBN 978-0-387-72828-5.
- Shilov, Georgi E. (1977), Linear algebra (translated and edited by Richard A. Silverman ed.), New York: Dover Publications, ISBN 0-486-63518-X.
- Hefferon, Jim (2001), Linear Algebra, Online book, St Michael's College, Colchester, Vermont, USA.
- Kuttler, Kenneth (2007), An introduction to linear algebra (PDF), Online e-book in PDF format, Brigham Young University.
- Demmel, James W. (1997), Applied numerical linear algebra, SIAM, ISBN 0-89871-389-7.
- Beezer, Robert A. (2006), A first course in linear algebra, Free online book under GNU licence, University of Puget Sound.
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External links
- MIT Video Lecture on Eigenvalues and Eigenvectors at Google Video, from MIT OpenCourseWare
- ARPACK is a collection of FORTRAN subroutines for solving large scale (sparse) eigenproblems.
- IRBLEIGS, has MATLAB code with similar capabilities to ARPACK. (See this paper for a comparison between IRBLEIGS and ARPACK.)
- LAPACK is a collection of FORTRAN subroutines for solving dense linear algebra problems
- ALGLIB includes a partial port of the LAPACK to C++, C#, Delphi, etc.
- Eigenvalue (of a matrix) on PlanetMath
- MathWorld: Eigenvector
- Online calculator for Eigenvalues and Eigenvectors
- Online Matrix Calculator Calculates eigenvalues, eigenvectors and other decompositions of matrices online
- Vanderplaats Research and Development - Provides the SMS eigenvalue solver for Structural Finite Element. The solver is in the GENESIS program as well as other commercial programs. SMS can be easily use with MSC.Nastran or NX/Nastran via DMAPs.
- What are Eigen Values? from PhysLink.com's "Ask the Experts"
- Templates for the Solution of Algebraic Eigenvalue Problems Edited by Zhaojun Bai, James Demmel, Jack Dongarra, Axel Ruhe, and Henk van der Vorst (a guide to the numerical solution of eigenvalue problems)
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