eigh(a, UPLO='L')
Returns two objects, a 1-D array containing the eigenvalues of a, and a 2-D square array or matrix (depending on the input type) of the corresponding eigenvectors (in columns).
Broadcasting rules apply, see the numpy.linalg
documentation for details.
The eigenvalues/eigenvectors are computed using LAPACK routines _syevd
, _heevd
.
The eigenvalues of real symmetric or complex Hermitian matrices are always real. The array v of (column) eigenvectors is unitary and a, w, and v satisfy the equations dot(a, v[:, i]) = w[i] * v[:, i]
.
Hermitian or real symmetric matrices whose eigenvalues and eigenvectors are to be computed.
Specifies whether the calculation is done with the lower triangular part of a ('L', default) or the upper triangular part ('U'). Irrespective of this value only the real parts of the diagonal will be considered in the computation to preserve the notion of a Hermitian matrix. It therefore follows that the imaginary part of the diagonal will always be treated as zero.
If the eigenvalue computation does not converge.
The eigenvalues in ascending order, each repeated according to its multiplicity.
The column v[:, i]
is the normalized eigenvector corresponding to the eigenvalue w[i]
. Will return a matrix object if a is a matrix object.
Return the eigenvalues and eigenvectors of a complex Hermitian (conjugate symmetric) or a real symmetric matrix.
eig
eigenvalues and right eigenvectors for non-symmetric arrays.
eigvals
eigenvalues of non-symmetric arrays.
eigvalsh
eigenvalues of real symmetric or complex Hermitian (conjugate symmetric) arrays.
scipy.linalg.eigh
Similar function in SciPy (but also solves the generalized eigenvalue problem).
>>> from numpy import linalg as LA
... a = np.array([[1, -2j], [2j, 5]])
... a array([[ 1.+0.j, -0.-2.j], [ 0.+2.j, 5.+0.j]])
>>> w, v = LA.eigh(a)
... w; v array([0.17157288, 5.82842712]) array([[-0.92387953+0.j , -0.38268343+0.j ], # may vary [ 0. +0.38268343j, 0. -0.92387953j]])
>>> np.dot(a, v[:, 0]) - w[0] * v[:, 0] # verify 1st e-val/vec pair array([5.55111512e-17+0.0000000e+00j, 0.00000000e+00+1.2490009e-16j])
>>> np.dot(a, v[:, 1]) - w[1] * v[:, 1] # verify 2nd e-val/vec pair array([0.+0.j, 0.+0.j])
>>> A = np.matrix(a) # what happens if input is a matrix object
... A matrix([[ 1.+0.j, -0.-2.j], [ 0.+2.j, 5.+0.j]])
>>> w, v = LA.eigh(A)
... w; v array([0.17157288, 5.82842712]) matrix([[-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]])
... a array([[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]])
... b array([[5.+0.j, 0.-2.j], [0.+2.j, 2.+0.j]])
>>> wa, va = LA.eigh(a)
... wb, vb = LA.eig(b)
... wa; wb array([1., 6.]) array([6.+0.j, 1.+0.j])
>>> va; vb array([[-0.4472136 +0.j , -0.89442719+0.j ], # may vary [ 0. +0.89442719j, 0. -0.4472136j ]]) array([[ 0.89442719+0.j , -0. +0.4472136j], [-0. +0.4472136j, 0.89442719+0.j ]])See :
The following pages refer to to this document either explicitly or contain code examples using this.
numpy.linalg.eigvals
numpy.linalg.eigvalsh
numpy.linalg.eig
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