- Eigendecomposition of a matrix
In the mathematical discipline of
linear algebra , eigendecomposition or sometimesspectral decomposition is thefactorization of a matrix into acanonical form , whereby the matrix is represented in terms of itseigenvalues andeigenvectors .Fundamental theory of matrix eigenvectors and eigenvalues
A vector v of dimension "N" is an eigenvector of a square ("N"×"N") matrix A
if and only if it satisfies the linear equation:where λ is a scalar, termed the eigenvalue corresponding to v. The above equation is called the eigenvalue equation or the eigenvalue problem.This yields an equation for the eigenvalues:We call "p"(λ) the
characteristic polynomial , and the equation, called the characteristic equation, is an "N"th order polynomial equation in the unknown λ. This equation will have "N"λ distinct solutions, where 1 ≤ "N"λ ≤ "N" . The set of solutions, "i.e." the eigenvalues, is sometimes called the spectrum of A.We can factor "p" as:where:
For each eigenvalue, λ"i", we have a specific eigenvalue equation:There will be 1 ≤ "m""i" ≤ "n""i"
linearly independent solutions to each eigenvalue equation. The "m""i" solutions are the eigenvectors associated with the eigenvalue λ"i". The integer "m""i" is termed the geometric multiplicity of λ"i". It is important to keep in mind that 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"v, can be calculated by summing the geometric multiplicities:The eigenvectors can be indexed by eigenvalues, "i.e." using a double index, with v"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 v"k", with "k" = 1, 2, ... , "N"v.Eigendecomposition of a matrix
Let A be a square ("N"×"N") matrix with "N"
linearly independent eigenvectors, Then A can be factorized as:where Q is the square ("N"×"N") matrix whose "i"th column is the eigenvector of A and Λ is thediagonal matrix whose diagonal elements are the corresponding eigenvalues, "i.e.", .The eigenvectors are usually normalized, but they need not be. A non-normalized set of eigenvectors, can also be used as the columns of Q. That this is true can be understood by noting that the magnitude of the eigenvectors in Q gets canceled in the decomposition by the presence of Q−1.
Matrix inverse via eigendecomposition
If matrix A can be eigendecomposed and if none of its eigenvalues are zero, then A is
nonsingular and its inverse is given by:Because Λ is a
diagonal matrix , its inverse is easy to calculate::Functional calculus
The eigendecomposition allows for much easier computation of power series of matrices. If "f"("x") is given by :then we know that :Because Λ is a
diagonal matrix , functions of Λ are very easy to calculate::The off-diagonal elements of "f"(Λ) are zero; that is, "f"(Λ) is also a diagonal matrix. Therefore, calculating "f"(A) reduces to just calculating the function on each of the eigenvalues .A similar technique works more generally with the
holomorphic functional calculus , using:from above. Once again, we find that:Decomposition for special matrices
ymmetric matrices
Any eigenvector basis for a real
symmetric matrix is orthogonal, and can always be made into an orthonormal basis. Thus a real symmetric matrix can be decomposed as:where Q is anorthogonal matrix , and Λ is real and diagonal.Normal matrices
Any eigenvector basis for a complex
normal matrix is also orthogonal, so a real symmetric matrix can be decomposed as:where U is aunitary matrix . Further, if A is Hermitian, the diagonal matrix Λ has only real values, and if A is unitary, Λ takes all its values on the complex unit circle.Useful facts
Useful facts regarding eigenvalues
*The product of the eigenvalues is equal to the
determinant of A:Note that each eigenvalue is raised to the power "ni", the algebraic multiplicity.
*The sum of the eigenvalues is equal to the trace of A:Note that each eigenvalue is multiplied by "ni", the algebraic multiplicity.
*If the eigenvalues of A are λ"i", and A is invertible, then the eigenvalues of A-1 are simply λ"i"-1.
*If the eigenvalues of A are λ"i", then the eigenvalues of "f"(A) are simply "f"(λ"i"), for anyholomorphic function "f".Useful facts regarding eigenvectors
*If A is (real)
symmetric , then "N"v="N", the eigenvectors are real, mutuallyorthogonal and provide a basis for .
*The eigenvectors of A-1 are the same as the eigenvectors of A
*The eigenvectors of "f"(A) are the same as the eigenvectors of AUseful facts regarding eigendecomposition
* A can be eigendecomposed if and only if:
*If "p"(λ) has no repeated roots, i.e. "N"λ="N", then A can be eigendecomposed.
*The statement "A can be eigendecomposed" does "not" imply that A has an inverse.
*The statement "A has an inverse" does "not" imply that A can be eigendecomposed.
Useful facts regarding matrix inverse
* can be inverted if and only if:
*If and , the inverse is given by:Numerical computations
Numerical computation of eigenvalues
Suppose that we want to compute the eigenvalues of a given matrix. If the matrix is small, we can compute them symbolically using the
characteristic polynomial . However, this is often impossible for larger matrices, in which case we must use a numerical method.In practice, eigenvalues of large matrices are not computed using the characteristic polynomial. Computing the polynomial becomes expensive in itself, and exact (symbolic) roots of a high-degree polynomial can be difficult to compute and express: the
Abel–Ruffini theorem implies that the roots of high-degree (5 and above) polynomials cannot in general be expressed simply using "n"th roots. Effective numerical algorithms for approximating roots of polynomials exist, but small errors in the eigenvalues can lead to large errors in the eigenvectors. Therefore, general algorithms to find eigenvectors and eigenvalues are iterative. The easiest method is thepower method : arandom vector is chosen and a sequence ofunit vector s is computed as:
This
sequence willalmost always converge to an eigenvector corresponding to the eigenvalue of greatest magnitude. This algorithm is simple, but not very useful by itself. However, popular methods such as theQR algorithm are based on it.Numerical computation of eigenvectors
Once the eigenvalues are computed, the eigenvectors can be calculated by solving the equation:using
Gaussian elimination or any other method for solving matrix equations.Additional topics
Generalized eigenspaces
Recall that the "geometric" multiplicity of an eigenvalue can be described as the dimension of the associated eigenspace, the nullspace of λI − "A". The algebraic multiplicity can also be thought of as a dimension: it is the dimension of the associated generalized eigenspace (1st sense), which is the nullspace of the matrix (λI − "A")"k" for "any sufficiently large k". That is, it is the space of generalized eigenvectors (1st sense), where a generalized eigenvector is any vector which "eventually" becomes 0 if λI − "A" is applied to it enough times successively. Any eigenvector is a generalized eigenvector, and so each eigenspace is contained in the associated generalized eigenspace. This provides an easy proof that the geometric multiplicity is always less than or equal to the algebraic multiplicity.
This usage should not be confused with the "generalized eigenvalue problem" described below.
Conjugate eigenvector
A conjugate eigenvector or coneigenvector is a vector sent after transformation to a scalar multiple of its conjugate, where the scalar is called the conjugate eigenvalue or coneigenvalue of the linear transformation. The coneigenvectors and coneigenvalues represent essentially the same information and meaning as the regular eigenvectors and eigenvalues, but arise when an alternative coordinate system is used. The corresponding equation is
:
For example, in coherent electromagnetic scattering theory, the linear transformation "A" represents the action performed by the scattering object, and the eigenvectors represent polarization states of the electromagnetic wave. In
optics , the coordinate system is defined from the wave's viewpoint, known as theForward Scattering Alignment (FSA), and gives rise to a regular eigenvalue equation, whereas inradar , the coordinate system is defined from the radar's viewpoint, known as theBack Scattering Alignment (BSA), and gives rise to a coneigenvalue equation.Generalized eigenvalue problem
A generalized eigenvalue problem (2nd sense) is of the form: where "A" and "B" are matrices. The generalized eigenvalues (2nd sense) λ can be obtained by solving the equation:The set of matrices of the form , where is a complex number, is called a "pencil".If "B" is invertible, then the original problem can be written in the form: which is a standard eigenvalue problem. However, in most situations it is preferable not to perform the inversion, but rather to solve the generalized eigenvalue problem as stated originally.
ee also
*
Matrix decomposition
*List of matrices
*Eigenvalue, eigenvector and eigenspace
*Spectral theorem Bibliography
* Golub, G. H. and Van Loan, C. F. (1996). "Matrix Computations". 3rd ed., Johns Hopkins University Press, Baltimore. ISBN 0-8018-5414-8.
* Horn, Roger A. and Johnson, Charles R (1985). "Matrix Analysis". Cambridge University Press. ISBN 0-521-38632-2.
* Horn, Roger A. and Johnson, Charles R (1991). "Topics in Matrix Analysis". Cambridge University Press. ISBN 0-521-46713-6.
* Strang G (1998). "Introduction to Linear Algebra". 3rd ed., Wellesley-Cambridge Press. ISBN 0-9614088-5-5.
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