Eigenvalues and eigenvectors
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Syntax
e = eig(A)
[V,D] =eig(A)
[V,D,W]= eig(A)
e = eig(A,B)
[V,D] =eig(A,B)
[V,D,W]= eig(A,B)
[___] = eig(A,balanceOption)
[___] = eig(A,B,algorithm)
[___] = eig(___,outputForm)
Description
example
e = eig(A)
returnsa column vector containing the eigenvalues of square matrix A
.
example
[V,D] =eig(A)
returns diagonal matrix D
ofeigenvalues and matrix V
whose columns are thecorresponding right eigenvectors, so that A*V = V*D
.
example
[V,D,W]= eig(A)
also returns full matrix W
whosecolumns are the corresponding left eigenvectors, so that W'*A= D*W'
.
The eigenvalue problem is to determine the solution to the equation Av = λv,where A is an n
-by-n
matrix, v isa column vector of length n
, and λ isa scalar. The values of λ that satisfy theequation are the eigenvalues. The corresponding values of v thatsatisfy the equation are the right eigenvectors. The left eigenvectors, w,satisfy the equation w’A = λw’.
example
e = eig(A,B)
returnsa column vector containing the generalized eigenvalues of square matrices A
and B
.
example
[V,D] =eig(A,B)
returnsdiagonal matrix D
of generalized eigenvalues andfull matrix V
whose columns are the correspondingright eigenvectors, so that A*V = B*V*D
.
[V,D,W]= eig(A,B)
alsoreturns full matrix W
whose columns are the correspondingleft eigenvectors, so that W'*A = D*W'*B
.
The generalized eigenvalue problem is to determine the solutionto the equation Av = λBv,where A and B are n
-by-n
matrices, v isa column vector of length n
, and λ isa scalar. The values of λ that satisfy theequation are the generalized eigenvalues. The corresponding valuesof v are the generalized right eigenvectors. Theleft eigenvectors, w, satisfy the equation w’A = λw’B.
[___] = eig(A,balanceOption)
, where balanceOption
is "nobalance"
, disables the preliminary balancing step in the algorithm. The default for balanceOption
is "balance"
, which enables balancing. The eig
function can return any of the output arguments in previous syntaxes.
example
[___] = eig(A,B,algorithm)
, where algorithm
is "chol"
, uses the Cholesky factorization of B
to compute the generalized eigenvalues. The default for algorithm
depends on the properties of A
and B
, but is "qz"
, which uses the QZ algorithm, when A
or B
are not symmetric.
example
[___] = eig(___,outputForm)
returns the eigenvalues in the form specified by outputForm
using any of the input or output arguments in previous syntaxes. Specify outputForm
as "vector"
to return the eigenvalues in a column vector or as "matrix"
to return the eigenvalues in a diagonal matrix.
Examples
collapse all
Eigenvalues of Matrix
Open Live Script
Use gallery
to create a symmetric positive definite matrix.
A = gallery("lehmer",4)
A = 4×4 1.0000 0.5000 0.3333 0.2500 0.5000 1.0000 0.6667 0.5000 0.3333 0.6667 1.0000 0.7500 0.2500 0.5000 0.7500 1.0000
Calculate the eigenvalues of A
. The result is a column vector.
e = eig(A)
e = 4×1 0.2078 0.4078 0.8482 2.5362
Alternatively, use outputForm
to return the eigenvalues in a diagonal matrix.
D = eig(A,"matrix")
D = 4×4 0.2078 0 0 0 0 0.4078 0 0 0 0 0.8482 0 0 0 0 2.5362
Eigenvalues and Eigenvectors of Matrix
Open Live Script
Use gallery
to create a circulant matrix.
A = gallery("circul",3)
A = 3×3 1 2 3 3 1 2 2 3 1
Calculate the eigenvalues and right eigenvectors of A
.
[V,D] = eig(A)
V = 3×3 complex -0.5774 + 0.0000i 0.5774 + 0.0000i 0.5774 + 0.0000i -0.5774 + 0.0000i -0.2887 - 0.5000i -0.2887 + 0.5000i -0.5774 + 0.0000i -0.2887 + 0.5000i -0.2887 - 0.5000i
D = 3×3 complex 6.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i -1.5000 + 0.8660i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i -1.5000 - 0.8660i
Verify that the results satisfy A*V = V*D
.
A*V - V*D
ans = 3×3 complex10-14 × -0.2665 + 0.0000i -0.0888 - 0.0111i -0.0888 + 0.0111i 0.0888 + 0.0000i 0.0000 + 0.0833i 0.0000 - 0.0833i -0.0444 + 0.0000i -0.1157 + 0.0666i -0.1157 - 0.0666i
Ideally, the eigenvalue decomposition satisfies the relationship. Since eig
performs the decomposition using floating-point computations, then A*V
can, at best, approach V*D
. In other words, A*V - V*D
is close to, but not exactly, 0
.
Sorted Eigenvalues and Eigenvectors
Open Live Script
By default eig
does not always return the eigenvalues and eigenvectors in sorted order. Use the sort
function to put the eigenvalues in ascending order and reorder the corresponding eigenvectors.
Calculate the eigenvalues and eigenvectors of a 5-by-5 magic square matrix.
A = 5×5 17 24 1 8 15 23 5 7 14 16 4 6 13 20 22 10 12 19 21 3 11 18 25 2 9
[V,D] = eig(A)
V = 5×5 -0.4472 0.0976 -0.6330 0.6780 -0.2619 -0.4472 0.3525 0.5895 0.3223 -0.1732 -0.4472 0.5501 -0.3915 -0.5501 0.3915 -0.4472 -0.3223 0.1732 -0.3525 -0.5895 -0.4472 -0.6780 0.2619 -0.0976 0.6330
D = 5×5 65.0000 0 0 0 0 0 -21.2768 0 0 0 0 0 -13.1263 0 0 0 0 0 21.2768 0 0 0 0 0 13.1263
The eigenvalues of A
are on the diagonal of D
. However, the eigenvalues are unsorted.
Extract the eigenvalues from the diagonal of D
using diag(D)
, then sort the resulting vector in ascending order. The second output from sort
returns a permutation vector of indices.
[d,ind] = sort(diag(D))
d = 5×1 -21.2768 -13.1263 13.1263 21.2768 65.0000
ind = 5×1 2 3 5 4 1
Use ind
to reorder the diagonal elements of D
. Since the eigenvalues in D
correspond to the eigenvectors in the columns of V
, you must also reorder the columns of V
using the same indices.
Ds = D(ind,ind)
Ds = 5×5 -21.2768 0 0 0 0 0 -13.1263 0 0 0 0 0 13.1263 0 0 0 0 0 21.2768 0 0 0 0 0 65.0000
Vs = V(:,ind)
Vs = 5×5 0.0976 -0.6330 -0.2619 0.6780 -0.4472 0.3525 0.5895 -0.1732 0.3223 -0.4472 0.5501 -0.3915 0.3915 -0.5501 -0.4472 -0.3223 0.1732 -0.5895 -0.3525 -0.4472 -0.6780 0.2619 0.6330 -0.0976 -0.4472
Both (V,D)
and (Vs,Ds)
produce the eigenvalue decomposition of A
. The results of A*V-V*D
and A*Vs-Vs*Ds
agree, up to round-off error.
e1 = norm(A*V-V*D);e2 = norm(A*Vs-Vs*Ds);e = abs(e1 - e2)
e = 0
Left Eigenvectors
Open Live Script
Create a 3-by-3 matrix.
A = [1 7 3; 2 9 12; 5 22 7];
Calculate the right eigenvectors, V
, the eigenvalues, D
, and the left eigenvectors, W
.
[V,D,W] = eig(A)
V = 3×3 -0.2610 -0.9734 0.1891 -0.5870 0.2281 -0.5816 -0.7663 -0.0198 0.7912
D = 3×3 25.5548 0 0 0 -0.5789 0 0 0 -7.9759
W = 3×3 -0.1791 -0.9587 -0.1881 -0.8127 0.0649 -0.7477 -0.5545 0.2768 0.6368
Verify that the results satisfy W'*A = D*W'
.
W'*A - D*W'
ans = 3×310-13 × -0.0444 -0.1066 -0.0888 -0.0011 0.0442 0.0333 0 0.0266 0.0178
Ideally, the eigenvalue decomposition satisfies the relationship. Since eig
performs the decomposition using floating-point computations, then W'*A
can, at best, approach D*W'
. In other words, W'*A - D*W'
is close to, but not exactly, 0
.
Eigenvalues of Nondiagonalizable (Defective) Matrix
Open Live Script
Create a 3-by-3 matrix.
A = [3 1 0; 0 3 1; 0 0 3];
Calculate the eigenvalues and right eigenvectors of A
.
[V,D] = eig(A)
V = 3×3 1.0000 -1.0000 1.0000 0 0.0000 -0.0000 0 0 0.0000
D = 3×3 3 0 0 0 3 0 0 0 3
A
has repeated eigenvalues and the eigenvectors are not independent. This means that A
is not diagonalizable and is, therefore, defective.
Verify that V
and D
satisfy the equation, A*V = V*D
, even though A
is defective.
A*V - V*D
ans = 3×310-15 × 0 0.8882 -0.8882 0 0 0.0000 0 0 0
Ideally, the eigenvalue decomposition satisfies the relationship. Since eig
performs the decomposition using floating-point computations, then A*V
can, at best, approach V*D
. In other words, A*V - V*D
is close to, but not exactly, 0
.
Generalized Eigenvalues
Open Live Script
Create two matrices, A
and B
, then solve the generalized eigenvalue problem for the eigenvalues and right eigenvectors of the pair (A,B)
.
A = [1/sqrt(2) 0; 0 1];B = [0 1; -1/sqrt(2) 0];[V,D]=eig(A,B)
V = 2×2 complex 1.0000 + 0.0000i 1.0000 + 0.0000i 0.0000 - 0.7071i 0.0000 + 0.7071i
D = 2×2 complex 0.0000 + 1.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 - 1.0000i
Verify that the results satisfy A*V = B*V*D
.
A*V - B*V*D
ans = 2×2 0 0 0 0
The residual error A*V - B*V*D
is exactly zero.
Generalized Eigenvalues Using QZ Algorithm for Badly Conditioned Matrices
Open Live Script
Create a badly conditioned symmetric matrix containing values close to machine precision.
format long eA = diag([10^-16, 10^-15])
A = 2×2 1.000000000000000e-16 0 0 1.000000000000000e-15
Calculate the generalized eigenvalues and a set of right eigenvectors using the default algorithm. In this case, the default algorithm is "chol"
.
[V1,D1] = eig(A,A)
V1 = 2×2 1.000000000000000e+08 0 0 3.162277660168380e+07
D1 = 2×2 9.999999999999999e-01 0 0 1.000000000000000e+00
Now, calculate the generalized eigenvalues and a set of right eigenvectors using the "qz"
algorithm.
[V2,D2] = eig(A,A,"qz")
V2 = 2×2 1 0 0 1
D2 = 2×2 1 0 0 1
Check how well the "chol"
result satisfies A*V1 = A*V1*D1
.
format shortA*V1 - A*V1*D1
ans = 2×210-23 × 0.1654 0 0 -0.6617
Now, check how well the "qz"
result satisfies A*V2 = A*V2*D2
.
A*V2 - A*V2*D2
ans = 2×2 0 0 0 0
When both matrices are symmetric, eig
uses the "chol"
algorithm by default. In this case, the QZ algorithm returns more accurate results.
Generalized Eigenvalues Where One Matrix Is Singular
Open Live Script
Create a 2-by-2 identity matrix, A
, and a singular matrix, B
.
A = eye(2);B = [3 6; 4 8];
If you attempt to calculate the generalized eigenvalues of the matrix with the command [V,D] = eig(B\A)
, then MATLAB® returns an error because B\A
produces Inf
values.
Instead, calculate the generalized eigenvalues and right eigenvectors by passing both matrices to the eig
function.
[V,D] = eig(A,B)
V = 2×2 -0.7500 -1.0000 -1.0000 0.5000
D = 2×2 0.0909 0 0 Inf
It is better to pass both matrices separately, and let eig
choose the best algorithm to solve the problem. In this case, eig(A,B)
returns a set of eigenvectors and at least one real eigenvalue, even though B
is not invertible.
Verify for the first eigenvalue and the first eigenvector.
eigval = D(1,1);eigvec = V(:,1);A*eigvec - eigval*B*eigvec
ans = 2×110-15 × 0.1110 0.2220
Ideally, the eigenvalue decomposition satisfies the relationship. Since the decomposition is performed using floating-point computations, then A*eigvec
can, at best, approach eigval*B*eigvec
, as it does in this case.
Input Arguments
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A
— Input matrix
square matrix
Input matrix, specified as a real or complex square matrix.
Data Types: double
| single
Complex Number Support: Yes
B
— Generalized eigenvalue problem input matrix
square matrix
Generalized eigenvalue problem input matrix, specified as asquare matrix of real or complex values. B
mustbe the same size as A
.
Data Types: double
| single
Complex Number Support: Yes
balanceOption
— Balance option
"balance"
(default) | "nobalance"
Balance option, specified as: "balance"
, which enables a preliminary balancing step, or "nobalance"
which disables it. In most cases, the balancing step improves the conditioning of A
to produce more accurate results. However, there are cases in which balancing produces incorrect results. Specify "nobalance"
when A
contains values whose scale differs dramatically. For example, if A
contains nonzero integers, as well as very small (near zero) values, then the balancing step might scale the small values to make them as significant as the integers and produce inaccurate results.
"balance"
is the default behavior. For more information about balancing, see balance.
algorithm
— Generalized eigenvalue algorithm
"chol"
(default) | "qz"
Generalized eigenvalue algorithm, specified as "chol"
or "qz"
, which selects the algorithm to use for calculating the generalized eigenvalues of a pair.
algorithm | Description |
---|---|
"chol" | Computes the generalized eigenvalues of A and B using the Cholesky factorization of B . If A is not symmetric (Hermitian) or if B is not symmetric (Hermitian) positive definite, eig uses the QZ algorithm instead. |
"qz" | Uses the QZ algorithm, also known as the generalized Schurdecomposition. This algorithm ignores the symmetry of A and B . |
In general, the two algorithms return the same result. The QZalgorithm can be more stable for certain problems, such as those involvingbadly conditioned matrices.
Regardless of the algorithm you specify, the eig
functionalways uses the QZ algorithm when A
or B
arenot symmetric.
outputForm
— Output format of eigenvalues
"vector"
| "matrix"
Output format of eigenvalues, specified as "vector"
or "matrix"
. This option allows you to specify whether the eigenvalues are returned in a column vector or a diagonal matrix. The default behavior varies according to the number of outputs specified:
If you specify one output, such as
e = eig(A)
, then the eigenvalues are returned as a column vector by default.If you specify two or three outputs, such as
[V,D] = eig(A)
, then the eigenvalues are returned as a diagonal matrix,D
, by default.
Example: D = eig(A,"matrix")
returns a diagonal matrix of eigenvalues with the one output syntax.
Output Arguments
collapse all
e
— Eigenvalues (returned as vector)
column vector
Eigenvalues, returned as a column vector containing the eigenvalues (or generalized eigenvalues of a pair) with multiplicity. Each eigenvalue e(k)
corresponds with the right eigenvector V(:,k)
and the left eigenvector W(:,k)
.
When
A
is real symmetric or complex Hermitian, the values ofe
that satisfy Av = λv are real.When
A
is real skew-symmetric or complex skew-Hermitian, the values ofe
that satisfy Av = λv are imaginary.
V
— Right eigenvectors
square matrix
Right eigenvectors, returned as a square matrix whose columnsare the right eigenvectors of A
or generalizedright eigenvectors of the pair, (A,B)
. The formand normalization of V
depends on the combinationof input arguments:
[V,D] = eig(A)
returns matrixV
,whose columns are the right eigenvectors ofA
suchthatA*V = V*D
. The eigenvectors inV
arenormalized so that the 2-norm of each is 1.If
A
is real symmetric, Hermitian, or skew-Hermitian, then the right eigenvectorsV
are orthonormal.[V,D] = eig(A,"nobalance")
also returns matrixV
. However, the 2-norm of each eigenvector is not necessarily 1.[V,D] = eig(A,B)
and[V,D] = eig(A,B,algorithm)
returnV
as a matrix whose columns are the generalized right eigenvectors that satisfyA*V = B*V*D
. The 2-norm of each eigenvector is not necessarily 1. In this case,D
contains the generalized eigenvalues of the pair,(A,B)
, along the main diagonal.When
eig
uses the"chol"
algorithm with symmetric (Hermitian)A
and symmetric (Hermitian) positive definiteB
, it normalizes the eigenvectors inV
so that theB
-norm of each is 1.
Different machines and releases of MATLAB® can produce different eigenvectors that are still numerically accurate:
For real eigenvectors, the sign of the eigenvectors can change.
For complex eigenvectors, the eigenvectors can be multiplied by any complex number of magnitude 1.
For a multiple eigenvalue, its eigenvectors can be recombined through linear combinations. For example, if Ax = λx and Ay = λy, then A(x+y) = λ(x+y), so x+y also is an eigenvector of A.
D
— Eigenvalues (returned as matrix)
diagonal matrix
Eigenvalues, returned as a diagonal matrix with the eigenvalues of A
on the main diagonal or the eigenvalues of the pair, (A,B)
, with multiplicity, on the main diagonal. Each eigenvalue D(k,k)
corresponds with the right eigenvector V(:,k)
and the left eigenvector W(:,k)
.
When
A
is real symmetric or complex Hermitian, the values ofD
that satisfy Av = λv are real.When
A
is real skew-symmetric or complex skew-Hermitian, the values ofD
that satisfy Av = λv are imaginary.
W
— Left eigenvectors
square matrix
Left eigenvectors, returned as a square matrix whose columnsare the left eigenvectors of A
or generalized lefteigenvectors of the pair, (A,B)
. The form and normalizationof W
depends on the combination of input arguments:
[V,D,W] = eig(A)
returns matrixW
, whose columns are the left eigenvectors ofA
such thatW'*A = D*W'
. The eigenvectors inW
are normalized so that the 2-norm of each is 1. IfA
is symmetric, thenW
is the same asV
.[V,D,W] = eig(A,"nobalance")
also returns matrixW
. However, the 2-norm of each eigenvector is not necessarily 1.[V,D,W] = eig(A,B)
and[V,D,W]= eig(A,B,algorithm)
returnsW
as a matrixwhose columns are the generalized left eigenvectors that satisfyW'*A= D*W'*B
. The 2-norm of each eigenvector is not necessarily1. In this case,D
contains the generalized eigenvaluesof the pair,(A,B)
, along the main diagonal.If
A
andB
are symmetric, thenW
is the same asV
.
Different machines and releases of MATLAB can produce different eigenvectors that are still numerically accurate:
For real eigenvectors, the sign of the eigenvectors can change.
For complex eigenvectors, the eigenvectors can be multiplied by any complex number of magnitude 1.
For a multiple eigenvalue, its eigenvectors can be recombined through linear combinations. For example, if Ax = λx and Ay = λy, then A(x+y) = λ(x+y), so x+y also is an eigenvector of A.
More About
collapse all
Symmetric Matrix
A square matrix,
A
, is symmetric if it is equal to its nonconjugate transpose,A = A.'
.In terms of the matrix elements, this means that
Since real matrices are unaffected by complex conjugation, a real matrix that is symmetric is also Hermitian. For example, the matrix
is both symmetric and Hermitian.
Skew-Symmetric Matrix
A square matrix,
A
, is skew-symmetric if it is equal to the negation of its nonconjugate transpose,A = -A.'
.In terms of the matrix elements, this means that
Since real matrices are unaffected by complex conjugation, a real matrix that is skew-symmetric is also skew-Hermitian. For example, the matrix
is both skew-symmetric and skew-Hermitian.
Hermitian Matrix
A square matrix,
A
, is Hermitian if it is equal to its complex conjugate transpose,A = A'
.In terms of the matrix elements,
The entries on the diagonal of a Hermitian matrix are always real. Because real matrices are unaffected by complex conjugation, a real matrix that is symmetric is also Hermitian. For example, this matrix is both symmetric and Hermitian.
The eigenvalues of a Hermitian matrix are real.
Skew-Hermitian Matrix
A square matrix,
A
, is skew-Hermitian if it is equal to the negation of its complex conjugate transpose,A = -A'
.In terms of the matrix elements, this means that
The entries on the diagonal of a skew-Hermitian matrix are always pure imaginary or zero. Since real matrices are unaffected by complex conjugation, a real matrix that is skew-symmetric is also skew-Hermitian. For example, the matrix
is both skew-Hermitian and skew-symmetric.
The eigenvalues of a skew-Hermitian matrix are purely imaginary or zero.
Tips
The
eig
function can calculatethe eigenvalues of sparse matrices that are real and symmetric. Tocalculate the eigenvectors of a sparse matrix, or to calculate theeigenvalues of a sparse matrix that is not real and symmetric, usethe eigs function.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
Usage notes and limitations:
V
might represent a different basis of eigenvectors. This representation means that the eigenvector calculated by the generated code might be different in C and C++ code than in MATLAB. The eigenvalues inD
might not be in the same order as in MATLAB. You can verify theV
andD
values by using the eigenvalue problem equationA*V = V*D
.If you specify the LAPACK library callback class, then the code generator supports these options:
The computation of left eigenvectors.
Outputs are complex.
Code generation does not support sparse matrix inputs for this function.
Thread-Based Environment
Run code in the background using MATLAB® backgroundPool
or accelerate code with Parallel Computing Toolbox™ ThreadPool
.
This function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.
GPU Arrays
Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.
The eig
function partially supports GPU arrays. Some syntaxes of the function run on a GPU when you specify the input data as a gpuArray (Parallel Computing Toolbox). Usage notes and limitations:
For the generalized case,
eig(A,B)
,A
andB
must be real symmetric or complex Hermitian. Additionally,B
must be positive definite.The QZ algorithm,
eig(A,B,"qz")
, is not supported.
For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).
Distributed Arrays
Partition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™.
Usage notes and limitations:
For the generalized case,
eig(A,B)
,A
andB
must be real symmetric or complex Hermitian. Additionally,B
must be positive definite.These syntaxes are not supported for full distributed arrays:
[__] = eig(A,B,"qz")
[V,D,W] = eig(A,B)
For more information, see Run MATLAB Functions with Distributed Arrays (Parallel Computing Toolbox).
Version History
Introduced before R2006a
expand all
R2021b: eig
returns NaN
for nonfinite inputs
eig
returns NaN
values when the input contains nonfinite values (Inf
or NaN
). Previously, eig
threw an error when the input contained nonfinite values.
R2021a: Improved algorithm for skew-Hermitian matrices
The algorithm for input matrices that are skew-Hermitian was improved. With the function call [V,D] = eig(A)
, where A
is skew-Hermitian, eig
now guarantees that the matrix of eigenvectors V
is unitary and the diagonal matrix of eigenvalues D
is purely imaginary.
See Also
eigs | polyeig | balance | condeig | cdf2rdf | hess | schur | qz
Topics
- Eigenvalues
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