Documentation

eig

Eigenvalues and eigenvectors

Description

example

e = eig(A) returns a column vector containing the eigenvalues of square matrix A.

example

[V,D] = eig(A) returns diagonal matrix D of eigenvalues and matrix V whose columns are the corresponding right eigenvectors, so that A*V = V*D.

example

[V,D,W] = eig(A) also returns full matrix W whose columns 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 is a column vector of length n, and λ is a scalar. The values of λ that satisfy the equation are the eigenvalues. The corresponding values of v that satisfy the equation are the right eigenvectors. The left eigenvectors, w, satisfy the equation wA = λw’.

example

e = eig(A,B) returns a column vector containing the generalized eigenvalues of square matrices A and B.

example

[V,D] = eig(A,B) returns diagonal matrix D of generalized eigenvalues and full matrix V whose columns are the corresponding right eigenvectors, so that A*V = B*V*D.

[V,D,W] = eig(A,B) also returns full matrix W whose columns are the corresponding left eigenvectors, so that W'*A = D*W'*B.

The generalized eigenvalue problem is to determine the solution to the equation Av = λBv, where A and B are n-by-n matrices, v is a column vector of length n, and λ is a scalar. The values of λ that satisfy the equation are the generalized eigenvalues. The corresponding values of v are the generalized right eigenvectors. The left eigenvectors, w, satisfy the equation wA = λwB.

[___] = 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 generally 'qz', which uses the QZ algorithm.

If A is Hermitian and B is Hermitian positive definite, then the default for algorithm is 'chol'.

example

[___] = eig(___,eigvalOption) returns the eigenvalues in the form specified by eigvalOption using any of the input or output arguments in previous syntaxes. Specify eigvalOption as 'vector' to return the eigenvalues in a column vector or as 'matrix' to return the eigenvalues in a diagonal matrix.

Examples

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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 eigvalOption 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

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 complex
10-14 ×

-0.2220 + 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  -0.1110 + 0.0666i  -0.1110 - 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.

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 = magic(5)
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 = 1.8933e-29

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 = 3×3
10-13 ×

-0.0266   -0.2132   -0.1243
0.0056   -0.0286   -0.0072
-0.0022         0   -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.

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 = 3×3
10-15 ×

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.

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.

Create a badly conditioned symmetric matrix containing values close to machine precision.

format long e
A = 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 short
A*V1 - A*V1*D1
ans = 2×2
10-23 ×

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.

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 ${B}^{-1}A$ 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 $Av=\lambda Bv$ for the first eigenvalue and the first eigenvector.

eigval = D(1,1);
eigvec = V(:,1);
A*eigvec - eigval*B*eigvec
ans = 2×1
10-15 ×

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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Input matrix, specified as a real or complex square matrix.

Data Types: double | single
Complex Number Support: Yes

Generalized eigenvalue problem input matrix, specified as a square matrix of real or complex values. B must be the same size as A.

Data Types: double | single
Complex Number Support: Yes

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.

Generalized eigenvalue algorithm, specified as 'chol' or 'qz', which selects the algorithm to use for calculating the generalized eigenvalues of a pair.

algorithmDescription
'chol'Computes the generalized eigenvalues of A and B using the Cholesky factorization of B.
'qz'Uses the QZ algorithm, also known as the generalized Schur decomposition. This algorithm ignores the symmetry of A and B.

In general, the two algorithms return the same result. The QZ algorithm can be more stable for certain problems, such as those involving badly conditioned matrices.

When you omit the algorithm argument, the eig function selects an algorithm based on the properties of A and B. It uses the 'chol' algorithm for symmetric (Hermitian) A and symmetric (Hermitian) positive definite B. Otherwise, it uses the 'qz' algorithm.

Regardless of the algorithm you specify, the eig function always uses the QZ algorithm when A or B are not symmetric.

Eigenvalue option, 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

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Eigenvalues, returned as a column vector containing the eigenvalues (or generalized eigenvalues of a pair) with multiplicity.

When A is real and symmetric or complex Hermitian, the values of e that satisfy Av = λv are real.

Right eigenvectors, returned as a square matrix whose columns are the right eigenvectors of A or generalized right eigenvectors of the pair, (A,B). The form and normalization of V depends on the combination of input arguments:

• [V,D] = eig(A) returns matrix V, whose columns are the right eigenvectors of A such that A*V = V*D. The eigenvectors in V are normalized so that the 2-norm of each is 1.

If A is real symmetric, then the right eigenvectors, V, are orthonormal.

• [V,D] = eig(A,'nobalance') also returns matrix V. However, the 2-norm of each eigenvector is not necessarily 1.

• [V,D] = eig(A,B) and [V,D] = eig(A,B,algorithm) returns V as a matrix whose columns are the generalized right eigenvectors that satisfy A*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 definite B, it normalizes the eigenvectors in V so that the B-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.

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.

When A is real and symmetric or complex Hermitian, the values of D that satisfy Av = λv are real.

Left eigenvectors, returned as a square matrix whose columns are the left eigenvectors of A or generalized left eigenvectors of the pair, (A,B). The form and normalization of W depends on the combination of input arguments:

• [V,D,W] = eig(A) returns matrix W, whose columns are the left eigenvectors of A such that W'*A = D*W'. The eigenvectors in W are normalized so that the 2-norm of each is 1. If A is symmetric, then W is the same as V.

• [V,D,W] = eig(A,'nobalance') also returns matrix W. 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) returns W as a matrix whose columns are the generalized left eigenvectors that satisfy W'*A = D*W'*B. 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.

If A and B are symmetric, then W is the same as V.

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.

Tips

• The eig function can calculate the eigenvalues of sparse matrices that are real and symmetric. To calculate the eigenvectors of a sparse matrix, or to calculate the eigenvalues of a sparse matrix that is not real and symmetric, use the eigs function.