numpy.linalg.eig¶

numpy.linalg.
eig
(a)[source]¶ 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 eigenvaluew[i]
.
 Raises
 LinAlgError
If the eigenvalue computation does not converge.
See also
eigvals
eigenvalues of a nonsymmetric array.
eigh
eigenvalues and eigenvectors of a real symmetric or complex Hermitian (conjugate symmetric) array.
eigvalsh
eigenvalues of a real symmetric or complex Hermitian (conjugate symmetric) array.
scipy.linalg.eig
Similar function in SciPy that also solves the generalized eigenvalue problem.
scipy.linalg.schur
Best choice for unitary and other nonHermitian normal matrices.
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
a @ v = w * v
. Thus, the arrays a, w, and v satisfy the equationsa @ v[:,i] = w[i] * v[:,i]
for .The array v of eigenvectors may not be of maximum rank, that is, some of the columns may be linearly dependent, although roundoff error may obscure that fact. If the eigenvalues are all different, then theoretically the eigenvectors are linearly independent and a can be diagonalized by a similarity transformation using v, i.e,
inv(v) @ a @ v
is diagonal.For nonHermitian normal matrices the SciPy function
scipy.linalg.schur
is preferred because the matrix v is guaranteed to be unitary, which is not the case when usingeig
. The Schur factorization produces an upper triangular matrix rather than a diagonal matrix, but for normal matrices only the diagonal of the upper triangular matrix is needed, the rest is roundoff error.Finally, it is emphasized that v consists of the right (as in righthand side) eigenvectors of a. A vector y satisfying
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 evalues and evectors.
>>> 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 evalues and evectors; note that the evalues 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 , 0.707106780.j ], [0. 0.70710678j, 0. +0.70710678j]])
Complexvalued matrix with real evalues (but complexvalued evectors); 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.+0.j, 0.+0.j]) array([[ 0. +0.70710678j, 0.70710678+0.j ], # may vary [ 0.70710678+0.j , 0. +0.70710678j]])
Be careful about roundoff error!
>>> a = np.array([[1 + 1e9, 0], [0, 1  1e9]]) >>> # Theor. evalues are 1 +/ 1e9 >>> w, v = LA.eig(a) >>> w; v array([1., 1.]) array([[1., 0.], [0., 1.]])