numpy.linalg.eig — NumPy v1.15 Manual (original) (raw)
Compute the eigenvalues and right eigenvectors of a square array.
| Parameters: | a : (…, M, M) array Matrices for which the eigenvalues and right eigenvectors will be computed |
|---|---|
| Returns: | w : (…, M) array The eigenvalues, each repeated according to its multiplicity. The eigenvalues are not necessarily ordered. The resulting array will be of complex type, unless the imaginary part is zero in which case it will be cast to a real type. When _a_is real the resulting eigenvalues will be real (0 imaginary part) or occur in conjugate pairs v : (…, M, M) array The normalized (unit “length”) eigenvectors, such that the column v[:,i] is the eigenvector corresponding to the eigenvalue w[i]. |
| Raises: | LinAlgError If the eigenvalue computation does not converge. |
See also
eigenvalues of a non-symmetric array.
eigenvalues and eigenvectors of a symmetric or Hermitian (conjugate symmetric) array.
eigenvalues of a symmetric or Hermitian (conjugate symmetric) array.
Notes
New in version 1.8.0.
Broadcasting rules apply, see the numpy.linalg documentation for details.
This is implemented using the _geev LAPACK routines which compute the eigenvalues and eigenvectors of general square arrays.
The number w is an eigenvalue of a if there exists a vector_v_ such that dot(a,v) = w * v. Thus, the arrays a, w, and_v_ satisfy the equations dot(a[:,:], v[:,i]) = w[i] * v[:,i]for i \in \{0,...,M-1\}.
The array v of eigenvectors may not be of maximum rank, that is, some of the columns may be linearly dependent, although round-off error may obscure that fact. If the eigenvalues are all different, then theoretically the eigenvectors are linearly independent. Likewise, the (complex-valued) matrix of eigenvectors v is unitary if the matrix a is normal, i.e., if dot(a, a.H) = dot(a.H, a), where a.H denotes the conjugate transpose of a.
Finally, it is emphasized that v consists of the right (as in right-hand side) eigenvectors of a. A vector y satisfyingdot(y.T, a) = z * y.T for some number z is called a _left_eigenvector of a, and, in general, the left and right eigenvectors of a matrix are not necessarily the (perhaps conjugate) transposes of each other.
References
G. Strang, Linear Algebra and Its Applications, 2nd Ed., Orlando, FL, Academic Press, Inc., 1980, Various pp.
Examples
from numpy import linalg as LA
(Almost) trivial example with real e-values and e-vectors.
w, v = LA.eig(np.diag((1, 2, 3))) w; v array([ 1., 2., 3.]) array([[ 1., 0., 0.], [ 0., 1., 0.], [ 0., 0., 1.]])
Real matrix possessing complex e-values and e-vectors; note that the e-values are complex conjugates of each other.
w, v = LA.eig(np.array([[1, -1], [1, 1]])) w; v array([ 1. + 1.j, 1. - 1.j]) array([[ 0.70710678+0.j , 0.70710678+0.j ], [ 0.00000000-0.70710678j, 0.00000000+0.70710678j]])
Complex-valued matrix with real e-values (but complex-valued e-vectors); note that a.conj().T = a, i.e., a is Hermitian.
a = np.array([[1, 1j], [-1j, 1]]) w, v = LA.eig(a) w; v array([ 2.00000000e+00+0.j, 5.98651912e-36+0.j]) # i.e., {2, 0} array([[ 0.00000000+0.70710678j, 0.70710678+0.j ], [ 0.70710678+0.j , 0.00000000+0.70710678j]])
Be careful about round-off error!
a = np.array([[1 + 1e-9, 0], [0, 1 - 1e-9]])
Theor. e-values are 1 +/- 1e-9
w, v = LA.eig(a) w; v array([ 1., 1.]) array([[ 1., 0.], [ 0., 1.]])