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eig(a, b=None, left=False, right=True, overwrite_a=False, overwrite_b=False, check_finite=True, homogeneous_eigvals=False)

Find eigenvalues w and right or left eigenvectors of a general matrix:

a   vr[:,i] = w[i]        b   vr[:,i]
a.H vl[:,i] = w[i].conj() b.H vl[:,i]

where .H is the Hermitian conjugation.

Parameters

a : (M, M) array_like

A complex or real matrix whose eigenvalues and eigenvectors will be computed.

b : (M, M) array_like, optional

Right-hand side matrix in a generalized eigenvalue problem. Default is None, identity matrix is assumed.

left : bool, optional

Whether to calculate and return left eigenvectors. Default is False.

right : bool, optional

Whether to calculate and return right eigenvectors. Default is True.

overwrite_a : bool, optional

Whether to overwrite a; may improve performance. Default is False.

overwrite_b : bool, optional

Whether to overwrite b; may improve performance. Default is False.

check_finite : bool, optional

Whether to check that the input matrices contain only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs.

homogeneous_eigvals : bool, optional

If True, return the eigenvalues in homogeneous coordinates. In this case w is a (2, M) array so that:

w[1,i] a vr[:,i] = w[0,i] b vr[:,i]

Default is False.

Raises

LinAlgError: If eigenvalue computation does not converge.

Returns

w : (M,) or (2, M) double or complex ndarray: The eigenvalues, each repeated according to its multiplicity. The shape is (M,) unless homogeneous_eigvals=True .
vl : (M, M) double or complex ndarray: The normalized left eigenvector corresponding to the eigenvalue w[i] is the column vl[:,i]. Only returned if left=True .
vr : (M, M) double or complex ndarray: The normalized right eigenvector corresponding to the eigenvalue w[i] is the column vr[:,i] . Only returned if right=True .

Solve an ordinary or generalized eigenvalue problem of a square matrix.

Examples

>>> from scipy import linalg
... a = np.array([[0., -1.], [1., 0.]])
... linalg.eigvals(a)
array([0.+1.j, 0.-1.j])

>>> b = np.array([[0., 1.], [1., 1.]])
... linalg.eigvals(a, b)
array([ 1.+0.j, -1.+0.j])

>>> a = np.array([[3., 0., 0.], [0., 8., 0.], [0., 0., 7.]])
... linalg.eigvals(a, homogeneous_eigvals=True)
array([[3.+0.j, 8.+0.j, 7.+0.j],
       [1.+0.j, 1.+0.j, 1.+0.j]])

>>> a = np.array([[0., -1.], [1., 0.]])
... linalg.eigvals(a) == linalg.eig(a)[0]
array([ True,  True])

>>> linalg.eig(a, left=True, right=False)[1] # normalized left eigenvector
array([[-0.70710678+0.j        , -0.70710678-0.j        ],
       [-0.        +0.70710678j, -0.        -0.70710678j]])

>>> linalg.eig(a, left=False, right=True)[1] # normalized right eigenvector
array([[0.70710678+0.j        , 0.70710678-0.j        ],
       [0.        -0.70710678j, 0.        +0.70710678j]])

See :

Back References

The following pages refer to to this document either explicitly or contain code examples using this.

scipy.linalg._decomp.eigvals_banded scipy.linalg._decomp.eigvals scipy.linalg._decomp.eigh scipy.linalg._decomp.eigh_tridiagonal scipy.linalg._decomp.eig scipy.linalg._decomp.eig_banded

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