Eigenvalues and eigenvectors

In linear algebra, an eigenvector (/ˈaɪɡənˌvɛktər/) 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 ${\displaystyle �}$, 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.^[1] Loosely speaking, in a multidimensional vector space, the eigenvector is not rotated.

Formal definition[edit]

If T is a linear transformation from a vector space V over a field F into itself and v is a nonzero vector in V, then v is an eigenvector of T if T(v) is a scalar multiple of v. This can be written as

$${\displaystyle �(\mathbf{�})=�\mathbf{�},}$$

where λ is a scalar in F, known as the eigenvalue, characteristic value, or characteristic root associated with v.

There is a direct correspondence between n-by-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.^[2][3]

If V is finite-dimensional, the above equation is equivalent to^[4]

$${\displaystyle �\mathbf{�}=�\mathbf{�}.}$$

where A is the matrix representation of T and u is the coordinate vector of v.

Overview[edit]

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'.^[5][6] 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

$${\displaystyle �(\mathbf{�})=�\mathbf{�},}$$

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.

In this shear mapping the red arrow changes direction, but the blue arrow does not. The blue arrow is an eigenvector of this shear mapping because it does not change direction, and since its length is unchanged, its eigenvalue is 1.

A 2×2 real and symmetric matrix representing a stretching and shearing of the plane. The eigenvectors of the matrix (red lines) are the two special directions such that every point on them will just slide on them.

The Mona Lisa example pictured here provides a simple illustration. Each point on the painting can be represented as a vector pointing from the center of the painting to that point. The linear transformation in this example is called a shear mapping. Points in the top half are moved to the right, and points in the bottom half are moved to the left, proportional to how far they are from the horizontal axis that goes through the middle of the painting. The vectors pointing to each point in the original image are therefore tilted right or left, and made longer or shorter by the transformation. Points along the horizontal axis do not move at all when this transformation is applied. Therefore, any vector that points directly to the right or left with no vertical component is an eigenvector of this transformation, because the mapping does not change its direction. Moreover, these eigenvectors all have an eigenvalue equal to one, because the mapping does not change their length either.

Linear transformations can take many different forms, mapping vectors in a variety of vector spaces, so the eigenvectors can also take many forms. For example, the linear transformation could be a differential operator like ${\displaystyle \frac{�}{��}}$, in which case the eigenvectors are functions called eigenfunctions that are scaled by that differential operator, such as

$${\displaystyle \frac{�}{��}{�}^{��}=�{�}^{��}.}$$

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

$${\displaystyle �\mathbf{�}=�\mathbf{�},}$$

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:

• The set of all eigenvectors of a linear transformation, each paired with its corresponding eigenvalue, is called the eigensystem of that transformation.^[7][8]
• The set of all eigenvectors of T corresponding to the same eigenvalue, together with the zero vector, is called an eigenspace, or the characteristic space of T associated with that eigenvalue.^[9]
• If a set of eigenvectors of T forms a basis of the domain of T, then this basis is called an eigenbasis.

History[edit]

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.^[a] Joseph-Louis Lagrange realized that the principal axes are the eigenvectors of the inertia matrix.^[10]

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.^[11] Cauchy also coined the term racine caractéristique (characteristic root), for what is now called eigenvalue; his term survives in characteristic equation.^[b]

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.^[12] 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.^[11] This was extended by Charles Hermite in 1855 to what are now called Hermitian matrices.^[13]

Around the same time, Francesco Brioschi proved that the eigenvalues of orthogonal matrices lie on the unit circle,^[11] and Alfred Clebsch found the corresponding result for skew-symmetric matrices.^[13] Finally, Karl Weierstrass clarified an important aspect in the stability theory started by Laplace, by realizing that defective matrices can cause instability.^[11]

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.^[14] 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.^[15]

At the start of the 20th century, David Hilbert studied the eigenvalues of integral operators by viewing the operators as infinite matrices.^[16] He was the first to use the German word eigen, which means "own",^[6] to denote eigenvalues and eigenvectors in 1904,^[c] though he may have been following a related usage by Hermann von Helmholtz. For some time, the standard term in English was "proper value", but the more distinctive term "eigenvalue" is the standard today.^[17]

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^[18] and Vera Kublanovskaya^[19] in 1961.^[20][21]

Eigenvalues and eigenvectors of matrices[edit]

See also: Euclidean vector and Matrix (mathematics)

Eigenvalues and eigenvectors are often introduced to students in the context of linear algebra courses focused on matrices.^[22][23] Furthermore, linear transformations over a finite-dimensional vector space can be represented using matrices,^[2][3] which is especially common in numerical and computational applications.^[24]

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

Consider n-dimensional vectors that are formed as a list of n scalars, such as the three-dimensional vectors

$${\displaystyle \mathbf{�}=\left[\begin{array}{c}1\\ -3\\ 4\end{array}\right]\text{and}\mathbf{�}=\left[\begin{array}{c}-20\\ 60\\ -80\end{array}\right].}$$

These vectors are said to be scalar multiples of each other, or parallel or collinear, if there is a scalar λ such that

$${\displaystyle \mathbf{�}=�\mathbf{�}.}$$

In this case ${\displaystyle �=-\frac{1}{20}}$.

Now consider the linear transformation of n-dimensional vectors defined by an n by n matrix A,

$${\displaystyle �\mathbf{�}=\mathbf{�},}$$

or

where, for each row,

$${\displaystyle {�}_{�}={�}_{�1}{�}_1+{�}_{�2}{�}_2+\cdots +{�}_{��}{�}_{�}=\sum _{�=1}^{�}{�}_{��}{�}_{�}.}$$

If it occurs that v and w are scalar multiples, that is if

${\displaystyle �\mathbf{�}=\mathbf{�}=�\mathbf{�},}$

(1)

then v is an eigenvector of the linear transformation A and the scale factor λ is the eigenvalue corresponding to that eigenvector. Equation (1) is the eigenvalue equation for the matrix A.

Equation (1) can be stated equivalently as

${\displaystyle \left(�-��\right)\mathbf{�}=\mathbf{0},}$

(2)

where I is the n by n identity matrix and 0 is the zero vector.

Eigenvalues and the characteristic polynomial[edit]

Main article: Characteristic polynomial

Equation (2) has a nonzero solution v if and only if the determinant of the matrix (A − λI) is zero. Therefore, the eigenvalues of A are values of λ that satisfy the equation

${\displaystyle |�-��|=0}$

(3)

Using the Leibniz formula for determinants, the left-hand side of Equation (3) 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)ⁿλⁿ. This polynomial is called the characteristic polynomial of A. Equation (3) 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,

${\displaystyle |�-��|=({�}_1-�)({�}_2-�)\cdots ({�}_{�}-�),}$

(4)

where each λ_i may be real but in general is a complex number. The numbers λ₁, λ₂, ..., 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 (A − λI), the characteristic polynomial of A is

Setting the characteristic polynomial equal to zero, it has roots at λ=1 and λ=3, 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 ${\displaystyle \left(�-��\right)\mathbf{�}=\mathbf{0}}$. In this example, the eigenvectors are any nonzero scalar multiples of

$${\displaystyle {\mathbf{�}}_{�=1}=\left[\begin{array}{c}1\\ -1\end{array}\right],{\mathbf{�}}_{�=3}=\left[\begin{array}{c}1\\ 1\end{array}\right].}$$

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.