Eigenvectors and eigenvalues of a matrix.

Syntax

V, D = eig(A)

V, D = eig(A, 'nobalance')

V, D = eig(A, B)

Argument

Name

Description

A

A square matrix (n-by-n); real or complex.

B

A square matrix (n-by-n); real or complex.

Outputs

Name

Description

V

Eigenvectors (n-by-n).

D

Eigenvalues stored as a diagonal matrix (n-by-n).

Example

Syntax

A = [2,1;3,4];

V1,D1 = eig(A)

print('V1 = ', V1)

print('D1 = ', D1)

print('A*V1 - V1*D1 = ',A*V1 - V1*D1);

VN,DN = eig(A, 'nobalance');

print('VN = ', VN)

print('DN = ', DN)

print('A*VN - VN*DN',A*VN - VN*DN)

A = [1, 2; 3, 8];

B = [8, 3; 4, 3];

V2, D2 = eig (A, B)

print('V2 = ', V2)

print('D2 = ', D2)

print('A*V2 - B*V2*D2 = ', A*V2 - B*V2*D2)

Result

V1 = [Matrix] 2 x 2

    -0.70711         -0.31623

    0.70711          -0.94868

D1 = [Matrix] 2 x 2

    1    0

    0    5

A*V1 - V1*D1 = [Matrix] 2 x 2

              -4.4409e-016        0

              -1.1102e-016        0

VN = [Matrix] 2 x 2

    -0.70711     -0.31623

    0.70711      -0.94868

DN = [Matrix] 2 x 2

    1      0

    0      5

A*VN - VN*DN = [Matrix] 2 x 2

              0     0

              0     0

V2 = [Matrix] 2 x 2

    -1         -0.32423

    0.36026    1

D2 = [Matrix] 2 x 2

    0.040392  0

    0         4.1263

A*V2 - B*V2*D2 = [Matrix] 2 x 2

                -2.7756e-016       -3.3307e-015

                2.6368e-016         8.8818e-016

Comments

V, D = eig (A) computes a diagonal matrix D (eigenvalues) and a full matrix V (eigenvectors) so that

A = V\V*D

or

A*V = V*D

or

Av = λv.

The matrix V is the modal matrix and the matrix D is the canonical form of A.

V, D = eig(A, nobalance) retrieves the eigenvectors and eigenvalues of A without balancing.

The nobalance argument has no affect on symmetric matrices.

V, D = eig (A, B) computes a diagonal matrix D of eigenvalues and a full matrix V whose columns are the corresponding eigenvectors as that A*V = B*V*D.

If A is a real matrix and symmetric, the eigenvalues and the eigenvectors are real.

If A is a real matrix and not symmetric, the eigenvalues and eigenvectors are complex.

If A is a complex matrix symmetric, the eigenvalues are real but the eigenvectors are complex.

If A is a complex matrix not symmetric, the eigenvalues and the eigenvectors are complex.

The eig function uses LAPACK functions (dsyev, zheev, dgeevx, zgeevx, dsygv, zhegv, dggev, zggev).

See Also:

svd