numpy.linalg.eigh#

linalg.eigh(a,UPLO='L')[source]#

Return the eigenvalues and eigenvectors of a complex Hermitian(conjugate symmetric) or a real symmetric matrix.

Returns two objects, a 1-D array containing the eigenvalues ofa, anda 2-D square array or matrix (depending on the input type) of thecorresponding eigenvectors (in columns).

Parameters:
a(…, M, M) array

Hermitian or real symmetric matrices whose eigenvalues andeigenvectors are to be computed.

UPLO{‘L’, ‘U’}, optional

Specifies whether the calculation is done with the lower triangularpart ofa (‘L’, default) or the upper triangular part (‘U’).Irrespective of this value only the real parts of the diagonal willbe considered in the computation to preserve the notion of a Hermitianmatrix. It therefore follows that the imaginary part of the diagonalwill always be treated as zero.

Returns:
A namedtuple with the following attributes:
eigenvalues(…, M) ndarray

The eigenvalues in ascending order, each repeated according toits multiplicity.

eigenvectors{(…, M, M) ndarray, (…, M, M) matrix}

The columneigenvectors[:,i] is the normalized eigenvectorcorresponding to the eigenvalueeigenvalues[i]. Will return amatrix object ifa is a matrix object.

Raises:
LinAlgError

If the eigenvalue computation does not converge.

See also

eigvalsh

eigenvalues of real symmetric or complex Hermitian (conjugate symmetric) arrays.

eig

eigenvalues and right eigenvectors for non-symmetric arrays.

eigvals

eigenvalues of non-symmetric arrays.

scipy.linalg.eigh

Similar function in SciPy (but also solves the generalized eigenvalue problem).

Notes

Broadcasting rules apply, see thenumpy.linalg documentation fordetails.

The eigenvalues/eigenvectors are computed using LAPACK routines_syevd,_heevd.

The eigenvalues of real symmetric or complex Hermitian matrices are alwaysreal.[1] The arrayeigenvalues of (column) eigenvectors is unitary anda,eigenvalues, andeigenvectors satisfy the equationsdot(a,eigenvectors[:,i])=eigenvalues[i]*eigenvectors[:,i].

References

[1]

G. Strang,Linear Algebra and Its Applications, 2nd Ed., Orlando,FL, Academic Press, Inc., 1980, pg. 222.

Examples

>>>importnumpyasnp>>>fromnumpyimportlinalgasLA>>>a=np.array([[1,-2j],[2j,5]])>>>aarray([[ 1.+0.j, -0.-2.j],       [ 0.+2.j,  5.+0.j]])>>>eigenvalues,eigenvectors=LA.eigh(a)>>>eigenvaluesarray([0.17157288, 5.82842712])>>>eigenvectorsarray([[-0.92387953+0.j        , -0.38268343+0.j        ], # may vary       [ 0.        +0.38268343j,  0.        -0.92387953j]])
>>>(np.dot(a,eigenvectors[:,0])-...eigenvalues[0]*eigenvectors[:,0])# verify 1st eigenval/vec pairarray([5.55111512e-17+0.0000000e+00j, 0.00000000e+00+1.2490009e-16j])>>>(np.dot(a,eigenvectors[:,1])-...eigenvalues[1]*eigenvectors[:,1])# verify 2nd eigenval/vec pairarray([0.+0.j, 0.+0.j])
>>>A=np.matrix(a)# what happens if input is a matrix object>>>Amatrix([[ 1.+0.j, -0.-2.j],        [ 0.+2.j,  5.+0.j]])>>>eigenvalues,eigenvectors=LA.eigh(A)>>>eigenvaluesarray([0.17157288, 5.82842712])>>>eigenvectorsmatrix([[-0.92387953+0.j        , -0.38268343+0.j        ], # may vary        [ 0.        +0.38268343j,  0.        -0.92387953j]])
>>># demonstrate the treatment of the imaginary part of the diagonal>>>a=np.array([[5+2j,9-2j],[0+2j,2-1j]])>>>aarray([[5.+2.j, 9.-2.j],       [0.+2.j, 2.-1.j]])>>># with UPLO='L' this is numerically equivalent to using LA.eig() with:>>>b=np.array([[5.+0.j,0.-2.j],[0.+2.j,2.-0.j]])>>>barray([[5.+0.j, 0.-2.j],       [0.+2.j, 2.+0.j]])>>>wa,va=LA.eigh(a)>>>wb,vb=LA.eig(b)>>>waarray([1., 6.])>>>wbarray([6.+0.j, 1.+0.j])>>>vaarray([[-0.4472136 +0.j        , -0.89442719+0.j        ], # may vary       [ 0.        +0.89442719j,  0.        -0.4472136j ]])>>>vbarray([[ 0.89442719+0.j       , -0.        +0.4472136j],       [-0.        +0.4472136j,  0.89442719+0.j       ]])
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