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 : array_like, shape (M, M)
UPLO : {‘L’, ‘U’}, optional
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Returns : | w : ndarray, shape (M,)
v : ndarray, or matrix object if a is, shape (M, M)
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Raises : | LinAlgError :
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See also
Notes
This is a simple interface to the LAPACK routines dsyevd and zheevd, which compute the eigenvalues and eigenvectors of real symmetric and complex Hermitian arrays, respectively.
The eigenvalues of real symmetric or complex Hermitian matrices are always real. [R37] 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
[R37] | (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]])