Return the eigenvalues and eigenvectors of a Hermitian or symmetric matrix.
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).
Parameters: | A : (..., M, M) array
UPLO : {‘L’, ‘U’}, optional
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Returns: | w : (..., M) ndarray
v : {(..., M, M) ndarray, (..., M, M) matrix}
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Raises: | LinAlgError :
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See also
Notes
Broadcasting rules apply, see the numpy.linalg documentation for details.
The eigenvalues/eigenvectors are computed using LAPACK routines _ssyevd, _heevd
The eigenvalues of real symmetric or complex Hermitian matrices are always real. [R38] The array v of (column) eigenvectors is unitary and a, w, and v satisfy the equations dot(a, v[:, i]) = w[i] * v[:, i].
References
[R38] | (1, 2) G. Strang, Linear Algebra and Its Applications, 2nd Ed., Orlando, FL, Academic Press, Inc., 1980, pg. 222. |
Examples
>>> 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 ],
[ 0.00000000+0.38268343j, 0.00000000-0.92387953j]])
>>> np.dot(a, v[:, 0]) - w[0] * v[:, 0] # verify 1st e-val/vec pair
array([2.77555756e-17 + 0.j, 0. + 1.38777878e-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 ],
[ 0.00000000+0.38268343j, 0.00000000-0.92387953j]])