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eig

Eigenvalues and eigenvectors

Syntax

  • [___] = eig(___,eigvalOption) example

Description

example

lambda = eig(A) returns a column vector containing the eigenvalues, with multiplicity, that satisfy 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.

example

lambda = eig(A,balanceOption) specifies a balancing option as one of two strings: 'balance', which enables a preliminary balancing step, or 'nobalance' which disables it.

example

lambda = eig(A,B) returns a vector containing the generalized eigenvalues of the pair, (A,B), that satisfy 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.

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lambda = eig(A,B,algorithm) specifies the generalized eigenvalue algorithm as one of two strings: 'qz', which uses the QZ algorithm, or 'chol', which uses the Cholesky factorization of B.

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[V,D] = eig(___) returns two optional outputs for any of the previous input syntaxes. D is a diagonal matrix containing the eigenvalues. V is a matrix whose columns are the corresponding right eigenvectors.

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[V,D,W] = eig(___) also returns W, a matrix whose columns are the corresponding left eigenvectors, using any of the previous input syntaxes. The left eigenvectors, w, satisfy the equation wA = λw.

example

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

Examples

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Eigenvalues of Real Symmetric Matrix

Use the gallery function to create symmetric positive definite matrix.

A = gallery('lehmer',4)
A =

    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.

lambda = eig(A)
lambda =

    0.2078
    0.4078
    0.8482
    2.5362

The result is a column vector.

Alternatively, use eigvalOption to return the eigenvalues in a diagonal matrix.

D = eig(A,'matrix')
D =

    0.2078         0         0         0
         0    0.4078         0         0
         0         0    0.8482         0
         0         0         0    2.5362

Eigenvalues and Eigenvectors of Nonsymmetric Matrix

Use the gallery function to create a circulant matrix.

A = gallery('circul',3)
A =

     1     2     3
     3     1     2
     2     3     1

Calculate the eigenvalues and right eigenvectors of A.

[V,D] = eig(A)
V =

  -0.5774 + 0.0000i   0.2887 - 0.5000i   0.2887 + 0.5000i
  -0.5774 + 0.0000i  -0.5774 + 0.0000i  -0.5774 + 0.0000i
  -0.5774 + 0.0000i   0.2887 + 0.5000i   0.2887 - 0.5000i

D =

   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 matrix V diagonalizes A, since the eigenvalues are distinct and the eigenvectors are independent.

V\A*V
ans =

   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

Eigenvalues and Eigenvectors of a Matrix Whose Elements Differ Dramatically in Scale

A = [ 3.0     -2.0      -0.9     2*eps;
     -2.0      4.0       1.0    -eps;
     -eps/4    eps/2    -1.0     0;
     -0.5     -0.5       0.1     1.0];

Calculate the eigenvalues and right eigenvectors using the default (balancing) behavior.

[VB,DB] = eig(A)
VB =

    0.6153   -0.4176   -0.0000   -0.1437
   -0.7881   -0.3261   -0.0000    0.1264
   -0.0000   -0.0000   -0.0000   -0.9196
    0.0189    0.8481    1.0000    0.3432


DB =

    5.5616         0         0         0
         0    1.4384         0         0
         0         0    1.0000         0
         0         0         0   -1.0000

Check to see if the results satisfy A*VB = VB*DB.

A*VB - VB*DB
ans =

    0.0000    0.0000   -0.0000    0.0000
         0   -0.0000    0.0000   -0.0000
    0.0000   -0.0000    0.0000    0.0000
         0    0.0000    0.0000    0.6031

This result does not satisfy A*VB = VB*DB. Ideally, the eigenvalue decomposition satisfies this relationship. Since MATLAB® 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.

Now, try calculating the eigenvalues and right eigenvectors without the balancing step.

[VN,DN] = eig(A,'nobalance')
VN =

    0.6153   -0.4176   -0.0000   -0.1528
   -0.7881   -0.3261         0    0.1345
   -0.0000   -0.0000   -0.0000   -0.9781
    0.0189    0.8481   -1.0000    0.0443


DN =

    5.5616         0         0         0
         0    1.4384         0         0
         0         0    1.0000         0
         0         0         0   -1.0000

Verify that the results satisfy A*VN = VN*DN.

A*VN - VN*DN
ans =

   1.0e-14 *

   -0.1776   -0.0111   -0.0559   -0.0167
    0.3553    0.1055    0.0336   -0.0194
    0.0017    0.0002    0.0007         0
    0.0264   -0.0222    0.0222    0.0097

A*VN - VN*DN is much closer to 0, so the 'nobalance' option produces more accurate results in this case.

Left Eigenvectors

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 =

   -0.2610   -0.9734    0.1891
   -0.5870    0.2281   -0.5816
   -0.7663   -0.0198    0.7912


D =

   25.5548         0         0
         0   -0.5789         0
         0         0   -7.9759


W =

   -0.1791   -0.9587   -0.1881
   -0.8127    0.0649   -0.7477
   -0.5545    0.2768    0.6368

Matrix W contains the left eigenvectors.

Verify the results satisfy W'*A = D*W'.

W'*A - D*W'
ans =

   1.0e-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 MATLAB 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 and Eigenvectors of Nondiagonalizable (Defective) Matrix

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 =

    1.0000   -1.0000    1.0000
         0    0.0000   -0.0000
         0         0    0.0000

D =

     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 =

   1.0e-15 *

         0    0.8882   -0.8882
         0         0    0.0000
         0         0         0

Ideally, the eigenvalue decomposition satisfies the relationship. Since MATLAB 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 and Eigenvectors

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 =

   1.0000 + 0.0000i   1.0000 + 0.0000i
   0.0000 - 0.7071i   0.0000 + 0.7071i

D =

   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 =

     0     0
     0     0

The residual error A*V - B*V*D is exactly zero.

Generalized Eigenvalues Using QZ Algorithm for Badly Conditioned Matrices

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

format long e
A = diag([10^-16, 10^-15])
A =

     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 =

     1.000000000000000e+08                         0
                         0     3.162277660168380e+07


D1 =

     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 =

     1     0
     0     1


D2 =

     1     0
     0     1

Check to see how well the 'chol' result satisfies A*V1 = A*V1*D1.

format
A*V1 - A*V1*D1
ans =

   1.0e-23 *

    0.1654         0
         0   -0.6617

Now, check to see how well the 'qz' result satisfies A*V2 = A*V2*D2.

A*V2 - A*V2*D2
ans =

     0     0
     0     0

When both matrices are Hermitian, eig uses the 'chol' algorithm by default. In this case, the QZ algorithm returns more accurate results.

Generalized Eigenvalues and Eigenvectors of a Pair in Which One Matrix is Singular

Create a 2-by-2 identity matrix, A, and a singular matrix, B.

A = eye(2);
B = [3 6; 4 8];

Try to calculate the generalized eigenvalues of the matrix, B-1A.

[V,D] = eig(B\A)
Warning: Matrix is singular to working precision. 
Error using eig
Input to EIG must not contain NaN or Inf.

Now calculate the generalized eigenvalues and right eigenvectors by passing both matrices to the eig function.

[V,D] = eig(A,B)
V =

   -0.7500   -1.0000
   -1.0000    0.5000

D =

    0.0909         0
         0       Inf

It's better to pass both matrices separately, and let eig choose the best algorithm to solve the problem. In this case, eig(A,B) returned a set of eigenvectors and at least one real eigenvalue, even though B is not invertible.

Verify Av = λBv for the first eigenvalue and the first eigenvector.

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

   1.0e-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 matrixsquare matrix

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

Data Types: double | single
Complex Number Support: Yes

B — Generalized eigenvalue problem input matrixsquare matrix

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

balanceOption — Balance option'balance' (default) | 'nobalance'

Balance option, specified as one two strings: '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. The eig function ignores balanceOption when A is symmetric.

Data Types: char

algorithm — Generalized eigenvalue algorithm'chol' | 'qz'

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.

eigvalOption — Eigenvalue option'vector' | 'matrix'

Eigenvalue option, specified as 'vector' or 'matrix'. This option allows you to specify whether the eigenvalues are returned in a column vector, lambda, or a diagonal matrix, D. The default behavior varies according to the number of outputs specified:

  • If you specify one output, such as lambda = eig(A), then the eigenvalues are returned as a column vector, lambda, 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.

Data Types: char

Output Arguments

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lambda — Eigenvaluescolumn vector

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 lambda that satisfy Av = λv are real.

  • When A is real and skew-symmetric or skew-Hermitian, the values of lambda that satisfy Av = λv are purely imaginary or zero.

V — Right eigenvectorssquare matrix

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.

  • [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.

    If A is symmetric and B is symmetric positive definite, then the eigenvectors in V are normalized so that the B-norm of each is 1.

D — Diagonal eigenvalues matrixdiagonal matrix

Diagonal eigenvalues matrix, returned as a matrix containing the eigenvalues of A or the generalized eigenvalues of the pair, (A,B), with multiplicity.

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

  • When A is real and skew-symmetric or skew-Hermitian, the values of D that satisfy Av = λv are purely imaginary or zero.

W — Left eigenvectorssquare matrix

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.

More About

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Tips

  • The eig function returns the eigenvalues of a real, symmetric, sparse matrix for the syntax, lambda = eig(A). Use the eigs function for all other sparse eigenvalue and eigenvector problems.

See Also

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