MATLAB Function Reference 
eig
Find eigenvalues and eigenvectors
Syntax
d = eig(A) d = eig(A,B) [V,D] = eig(A) [V,D] = eig(A,'nobalance') [V,D] = eig(A,B) [V,D] = eig(A,B,
flag
)
Description
d = eig(A)
returns a vector of the eigenvalues of matrix A
.
d = eig(A,B)
returns a vector containing the generalized eigenvalues, if A
and B
are square matrices.
Note If S is sparse and symmetric, you can use d = eig(S) to returns the eigenvalues of S . If S is sparse but not symmetric, or if you want to return the eigenvectors of S , use the function eigs instead of eig . 
[V,D] = eig(A)
produces matrices of eigenvalues (D
) and eigenvectors (V
) of matrix A
, so that A*V
=
V*D
. Matrix D
is the canonical form of A
–a diagonal matrix with A
‘s eigenvalues on the main diagonal. Matrix V
is the modal matrix–its columns are the eigenvectors of A
.
If W
is a matrix such that W'*A = D*W'
, the columns of W
are the left eigenvectors of A
. Use [W,D] = eig(A.'); W = conj(W)
to compute the left eigenvectors.
[V,D] = eig(A,'nobalance')
finds eigenvalues and eigenvectors without a preliminary balancing step. Ordinarily, balancing improves the conditioning of the input matrix, enabling more accurate computation of the eigenvectors and eigenvalues. However, if a matrix contains small elements that are really due to roundoff error, balancing may scale them up to make them as significant as the other elements of the original matrix, leading to incorrect eigenvectors. Use the nobalance
option in this event. See the balance
function for more details.
[V,D] = eig(A,B)
produces a diagonal matrix D
of generalized eigenvalues and a full matrix V
whose columns are the corresponding eigenvectors so that A*V
=
B*V*D
.
[V,D] = eig(A,B,
specifies the algorithm used to compute eigenvalues and eigenvectors. flag
)flag
can be:
'chol'  Computes the generalized eigenvalues of A and B using the Cholesky factorization of B . This is the default for symmetric (Hermitian) A and symmetric (Hermitian) positive definite B . 
'qz'  Ignores the symmetry, if any, and uses the QZ algorithm as it would for nonsymmetric (nonHermitian) A and B . 
Note For eig(A) , the eigenvectors are scaled so that the norm of each is 1.0. For eig(A,B) , eig(A,'nobalance') , and eig(A,B,flag) , the eigenvectors are not normalized. 
Remarks
The eigenvalue problem is to determine the nontrivial solutions of the equation
where is an n
byn
matrix, is a length n
column vector, and is a scalar. The n
values of that satisfy the equation are the eigenvalues, and the corresponding values of are the right eigenvectors. In MATLAB, the function eig
solves for the eigenvalues , and optionally the eigenvectors .
The generalized eigenvalue problem is to determine the nontrivial solutions of the equation
where both and are n
byn
matrices and is a scalar. The values of that satisfy the equation are the generalized eigenvalues and the corresponding values of are the generalized right eigenvectors.
If is nonsingular, the problem could be solved by reducing it to a standard eigenvalue problem
Because can be singular, an alternative algorithm, called the QZ method, is necessary.
When a matrix has no repeated eigenvalues, the eigenvectors are always independent and the eigenvector matrix V
diagonalizes the original matrix A
if applied as a similarity transformation. However, if a matrix has repeated eigenvalues, it is not similar to a diagonal matrix unless it has a full (independent) set of eigenvectors. If the eigenvectors are not independent then the original matrix is said to be defective. Even if a matrix is defective, the solution from eig
satisfies A*X
=
X*D
.
Examples
The matrix
B = [ 3 2 .9 2*eps
2 4 1 eps
eps/4 eps/2 1 0
.5 .5 .1 1 ];
has elements on the order of roundoff error. It is an example for which the nobalance
option is necessary to compute the eigenvectors correctly. Try the statements
[VB,DB] = eig(B) B*VB  VB*DB [VN,DN] = eig(B,'nobalance') B*VN  VN*DN
Algorithm
Inputs of Type Double
For inputs of type double
, MATLAB uses the following LAPACK routines to compute eigenvalues and eigenvectors.
Case  Routine 
Real symmetric A  DSYEV 
Real nonsymmetric A: 

 DGEEV (with the scaling factor SCLFAC = 2 in DGEBAL , instead of the LAPACK default value of 8 ) 
 DGEHRD , DHSEQR 
 DGEHRD , DORGHR , DHSEQR , DTREVC 
Hermitian A  ZHEEV 
NonHermitian A:  
 ZGEEV (with SCLFAC = 2 instead of 8 in ZGEBAL ) 
 ZGEHRD , ZHSEQR 
 ZGEHRD , ZUNGHR , ZHSEQR , ZTREVC 
Real symmetric A , symmetric positive definite B .  DSYGV 
Special case:eig(A,B,'qz') for real A , B (same as real nonsymmetric A , realgeneral B ) 

Real nonsymmetric A , real general B  DGGEV 
Complex Hermitian A, Hermitian positive definite B .  ZHEGV 
Special case: eig(A,B,'qz') for complex A or B (same as complex nonHermitian A, complex B ) 

Complex nonHermitian A, complex B  ZGGEV 
Inputs of Type Single
For inputs of type single
, MATLAB uses the following LAPACK routines to compute eigenvalues and eigenvectors.
Case  Routine 
Real symmetric A  SSYEV 
Real nonsymmetric A: 

 SGEEV (with the scaling factor SCLFAC = 2 in SGEBAL , instead of the LAPACK default value of 8 ) 
 SGEHRD , SHSEQR 
 SGEHRD , SORGHR , SHSEQR , STREVC 
Hermitian A  CHEEV 
NonHermitian A:  
 CGEEV 
 CGEHRD , CHSEQR 
 CGEHRD , CUNGHR , CHSEQR , CTREVC 
Real symmetric A , symmetric positive definite B .  CSYGV 
Special case:eig(A,B,'qz') for real A , B (same as real nonsymmetric A , realgeneral B ) 

Real nonsymmetric A , real general B  SGGEV 
Complex Hermitian A, Hermitian positive definite B .  CHEGV 
Special case: eig(A,B,'qz') for complex A or B (same as complex nonHermitian A, complex B ) 

Complex nonHermitian A, complex B  CGGEV 
See Also
balance
, condeig
, eigs
, hess
, qz
, schur
References
[1] Anderson, E., Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra,
J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen,
LAPACK User’s Guide ( http://www.netlib.org/lapack/lug/
lapack_lug.html ), Third Edition, SIAM, Philadelphia, 1999.
edit  eigs 
© 19942005 The MathWorks, Inc.
MATLAB Function Reference 
eig
Find eigenvalues and eigenvectors
Syntax
d = eig(A) d = eig(A,B) [V,D] = eig(A) [V,D] = eig(A,'nobalance') [V,D] = eig(A,B) [V,D] = eig(A,B,
flag
)
Description
d = eig(A)
returns a vector of the eigenvalues of matrix A
.
d = eig(A,B)
returns a vector containing the generalized eigenvalues, if A
and B
are square matrices.
Note If S is sparse and symmetric, you can use d = eig(S) to returns the eigenvalues of S . If S is sparse but not symmetric, or if you want to return the eigenvectors of S , use the function eigs instead of eig . 
[V,D] = eig(A)
produces matrices of eigenvalues (D
) and eigenvectors (V
) of matrix A
, so that A*V
=
V*D
. Matrix D
is the canonical form of A
–a diagonal matrix with A
‘s eigenvalues on the main diagonal. Matrix V
is the modal matrix–its columns are the eigenvectors of A
.
If W
is a matrix such that W'*A = D*W'
, the columns of W
are the left eigenvectors of A
. Use [W,D] = eig(A.'); W = conj(W)
to compute the left eigenvectors.
[V,D] = eig(A,'nobalance')
finds eigenvalues and eigenvectors without a preliminary balancing step. Ordinarily, balancing improves the conditioning of the input matrix, enabling more accurate computation of the eigenvectors and eigenvalues. However, if a matrix contains small elements that are really due to roundoff error, balancing may scale them up to make them as significant as the other elements of the original matrix, leading to incorrect eigenvectors. Use the nobalance
option in this event. See the balance
function for more details.
[V,D] = eig(A,B)
produces a diagonal matrix D
of generalized eigenvalues and a full matrix V
whose columns are the corresponding eigenvectors so that A*V
=
B*V*D
.
[V,D] = eig(A,B,
specifies the algorithm used to compute eigenvalues and eigenvectors. flag
)flag
can be:
'chol'  Computes the generalized eigenvalues of A and B using the Cholesky factorization of B . This is the default for symmetric (Hermitian) A and symmetric (Hermitian) positive definite B . 
'qz'  Ignores the symmetry, if any, and uses the QZ algorithm as it would for nonsymmetric (nonHermitian) A and B . 
Note For eig(A) , the eigenvectors are scaled so that the norm of each is 1.0. For eig(A,B) , eig(A,'nobalance') , and eig(A,B,flag) , the eigenvectors are not normalized. 
Remarks
The eigenvalue problem is to determine the nontrivial solutions of the equation
where is an n
byn
matrix, is a length n
column vector, and is a scalar. The n
values of that satisfy the equation are the eigenvalues, and the corresponding values of are the right eigenvectors. In MATLAB, the function eig
solves for the eigenvalues , and optionally the eigenvectors .
The generalized eigenvalue problem is to determine the nontrivial solutions of the equation
where both and are n
byn
matrices and is a scalar. The values of that satisfy the equation are the generalized eigenvalues and the corresponding values of are the generalized right eigenvectors.
If is nonsingular, the problem could be solved by reducing it to a standard eigenvalue problem
Because can be singular, an alternative algorithm, called the QZ method, is necessary.
When a matrix has no repeated eigenvalues, the eigenvectors are always independent and the eigenvector matrix V
diagonalizes the original matrix A
if applied as a similarity transformation. However, if a matrix has repeated eigenvalues, it is not similar to a diagonal matrix unless it has a full (independent) set of eigenvectors. If the eigenvectors are not independent then the original matrix is said to be defective. Even if a matrix is defective, the solution from eig
satisfies A*X
=
X*D
.
Examples
The matrix
B = [ 3 2 .9 2*eps
2 4 1 eps
eps/4 eps/2 1 0
.5 .5 .1 1 ];
has elements on the order of roundoff error. It is an example for which the nobalance
option is necessary to compute the eigenvectors correctly. Try the statements
[VB,DB] = eig(B) B*VB  VB*DB [VN,DN] = eig(B,'nobalance') B*VN  VN*DN
Algorithm
Inputs of Type Double
For inputs of type double
, MATLAB uses the following LAPACK routines to compute eigenvalues and eigenvectors.
Case  Routine 
Real symmetric A  DSYEV 
Real nonsymmetric A: 

 DGEEV (with the scaling factor SCLFAC = 2 in DGEBAL , instead of the LAPACK default value of 8 ) 
 DGEHRD , DHSEQR 
 DGEHRD , DORGHR , DHSEQR , DTREVC 
Hermitian A  ZHEEV 
NonHermitian A:  
 ZGEEV (with SCLFAC = 2 instead of 8 in ZGEBAL ) 
 ZGEHRD , ZHSEQR 
 ZGEHRD , ZUNGHR , ZHSEQR , ZTREVC 
Real symmetric A , symmetric positive definite B .  DSYGV 
Special case:eig(A,B,'qz') for real A , B (same as real nonsymmetric A , realgeneral B ) 

Real nonsymmetric A , real general B  DGGEV 
Complex Hermitian A, Hermitian positive definite B .  ZHEGV 
Special case: eig(A,B,'qz') for complex A or B (same as complex nonHermitian A, complex B ) 

Complex nonHermitian A, complex B  ZGGEV 
Inputs of Type Single
For inputs of type single
, MATLAB uses the following LAPACK routines to compute eigenvalues and eigenvectors.
Case  Routine 
Real symmetric A  SSYEV 
Real nonsymmetric A: 

 SGEEV (with the scaling factor SCLFAC = 2 in SGEBAL , instead of the LAPACK default value of 8 ) 
 SGEHRD , SHSEQR 
 SGEHRD , SORGHR , SHSEQR , STREVC 
Hermitian A  CHEEV 
NonHermitian A:  
 CGEEV 
 CGEHRD , CHSEQR 
 CGEHRD , CUNGHR , CHSEQR , CTREVC 
Real symmetric A , symmetric positive definite B .  CSYGV 
Special case:eig(A,B,'qz') for real A , B (same as real nonsymmetric A , realgeneral B ) 

Real nonsymmetric A , real general B  SGGEV 
Complex Hermitian A, Hermitian positive definite B .  CHEGV 
Special case: eig(A,B,'qz') for complex A or B (same as complex nonHermitian A, complex B ) 

Complex nonHermitian A, complex B  CGGEV 
See Also
balance
, condeig
, eigs
, hess
, qz
, schur
References
[1] Anderson, E., Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra,
J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen,
LAPACK User’s Guide ( http://www.netlib.org/lapack/lug/
lapack_lug.html ), Third Edition, SIAM, Philadelphia, 1999.
edit  eigs 
© 19942005 The MathWorks, Inc.