numpy.linalg.eigvalsh#
Compute the eigenvalues of a complex Hermitian or real symmetric matrix.
Main difference from eigh: the eigenvectors are not computed.
Parameters : a (…, M, M) array_like
A complex- or real-valued matrix whose eigenvalues are to be computed.
UPLO , optional
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.
Returns : w (…, M,) ndarray
The eigenvalues in ascending order, each repeated according to its multiplicity.
If the eigenvalue computation does not converge.
eigenvalues and eigenvectors of real symmetric or complex Hermitian (conjugate symmetric) arrays.
eigenvalues of general real or complex arrays.
eigenvalues and right eigenvectors of general real or complex arrays.
Similar function in SciPy.
Broadcasting rules apply, see the numpy.linalg documentation for details.
The eigenvalues are computed using LAPACK routines _syevd , _heevd .
>>> from numpy import linalg as LA >>> a = np.array([[1, -2j], [2j, 5]]) >>> LA.eigvalsh(a) array([ 0.17157288, 5.82842712]) # may vary
>>> # 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.eigvals() >>> # 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 = LA.eigvalsh(a) >>> wb = LA.eigvals(b) >>> wa; wb array([1., 6.]) array([6.+0.j, 1.+0.j])
numpy.linalg.eig#
Compute the eigenvalues and right eigenvectors of a square array.
Parameters : a (…, M, M) array
Matrices for which the eigenvalues and right eigenvectors will be computed
Returns : A namedtuple with the following attributes: eigenvalues (…, M) array
The eigenvalues, each repeated according to its multiplicity. The eigenvalues are not necessarily ordered. The resulting array will be of complex type, unless the imaginary part is zero in which case it will be cast to a real type. When a is real the resulting eigenvalues will be real (0 imaginary part) or occur in conjugate pairs
eigenvectors (…, M, M) array
The normalized (unit “length”) eigenvectors, such that the column eigenvectors[:,i] is the eigenvector corresponding to the eigenvalue eigenvalues[i] .
If the eigenvalue computation does not converge.
eigenvalues of a non-symmetric array.
eigenvalues and eigenvectors of a real symmetric or complex Hermitian (conjugate symmetric) array.
eigenvalues of a real symmetric or complex Hermitian (conjugate symmetric) array.
Similar function in SciPy that also solves the generalized eigenvalue problem.
Best choice for unitary and other non-Hermitian normal matrices.
Broadcasting rules apply, see the numpy.linalg documentation for details.
This is implemented using the _geev LAPACK routines which compute the eigenvalues and eigenvectors of general square arrays.
The number w is an eigenvalue of a if there exists a vector v such that a @ v = w * v . Thus, the arrays a, eigenvalues, and eigenvectors satisfy the equations a @ eigenvectors[:,i] = eigenvalues[i] * eigenvalues[:,i] for \(i \in \\) .
The array eigenvectors may not be of maximum rank, that is, some of the columns may be linearly dependent, although round-off error may obscure that fact. If the eigenvalues are all different, then theoretically the eigenvectors are linearly independent and a can be diagonalized by a similarity transformation using eigenvectors, i.e, inv(eigenvectors) @ a @ eigenvectors is diagonal.
For non-Hermitian normal matrices the SciPy function scipy.linalg.schur is preferred because the matrix eigenvectors is guaranteed to be unitary, which is not the case when using eig . The Schur factorization produces an upper triangular matrix rather than a diagonal matrix, but for normal matrices only the diagonal of the upper triangular matrix is needed, the rest is roundoff error.
Finally, it is emphasized that eigenvectors consists of the right (as in right-hand side) eigenvectors of a. A vector y satisfying y.T @ a = z * y.T for some number z is called a left eigenvector of a, and, in general, the left and right eigenvectors of a matrix are not necessarily the (perhaps conjugate) transposes of each other.
G. Strang, Linear Algebra and Its Applications, 2nd Ed., Orlando, FL, Academic Press, Inc., 1980, Various pp.
>>> from numpy import linalg as LA
(Almost) trivial example with real eigenvalues and eigenvectors.
>>> eigenvalues, eigenvectors = LA.eig(np.diag((1, 2, 3))) >>> eigenvalues array([1., 2., 3.]) >>> eigenvectors array([[1., 0., 0.], [0., 1., 0.], [0., 0., 1.]])
Real matrix possessing complex eigenvalues and eigenvectors; note that the eigenvalues are complex conjugates of each other.
>>> eigenvalues, eigenvectors = LA.eig(np.array([[1, -1], [1, 1]])) >>> eigenvalues array([1.+1.j, 1.-1.j]) >>> eigenvectors array([[0.70710678+0.j , 0.70710678-0.j ], [0. -0.70710678j, 0. +0.70710678j]])
Complex-valued matrix with real eigenvalues (but complex-valued eigenvectors); note that a.conj().T == a , i.e., a is Hermitian.
>>> a = np.array([[1, 1j], [-1j, 1]]) >>> eigenvalues, eigenvectors = LA.eig(a) >>> eigenvalues array([2.+0.j, 0.+0.j]) >>> eigenvectors array([[ 0. +0.70710678j, 0.70710678+0.j ], # may vary [ 0.70710678+0.j , -0. +0.70710678j]])
Be careful about round-off error!
>>> a = np.array([[1 + 1e-9, 0], [0, 1 - 1e-9]]) >>> # Theor. eigenvalues are 1 +/- 1e-9 >>> eigenvalues, eigenvectors = LA.eig(a) >>> eigenvalues array([1., 1.]) >>> eigenvectors array([[1., 0.], [0., 1.]])
numpy.linalg.eigh#
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 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
Hermitian or real symmetric matrices whose eigenvalues and eigenvectors are to be computed.
UPLO , optional
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.
Returns : A namedtuple with the following attributes: eigenvalues (…, M) ndarray
The eigenvalues in ascending order, each repeated according to its multiplicity.
eigenvectors
The column eigenvectors[:, i] is the normalized eigenvector corresponding to the eigenvalue eigenvalues[i] . Will return a matrix object if a is a matrix object.
If the eigenvalue computation does not converge.
eigenvalues of real symmetric or complex Hermitian (conjugate symmetric) arrays.
eigenvalues and right eigenvectors for non-symmetric arrays.
eigenvalues of non-symmetric arrays.
Similar function in SciPy (but also solves the generalized eigenvalue problem).
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. [1] The array eigenvalues of (column) eigenvectors is unitary and a, eigenvalues, and eigenvectors satisfy the equations dot(a, eigenvectors[:, i]) = eigenvalues[i] * eigenvectors[:, i] .
G. Strang, Linear Algebra and Its Applications, 2nd Ed., Orlando, FL, Academic Press, Inc., 1980, pg. 222.
>>> 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]]) >>> eigenvalues, eigenvectors = LA.eigh(a) >>> eigenvalues array([0.17157288, 5.82842712]) >>> eigenvectors array([[-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 pair array([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 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]]) >>> eigenvalues, eigenvectors = LA.eigh(A) >>> eigenvalues array([0.17157288, 5.82842712]) >>> eigenvectors 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 ]])
numpy.linalg.eigvals#
Main difference between eigvals and eig : the eigenvectors aren’t returned.
Parameters : a (…, M, M) array_like
A complex- or real-valued matrix whose eigenvalues will be computed.
Returns : w (…, M,) ndarray
The eigenvalues, each repeated according to its multiplicity. They are not necessarily ordered, nor are they necessarily real for real matrices.
If the eigenvalue computation does not converge.
eigenvalues and right eigenvectors of general arrays
eigenvalues of real symmetric or complex Hermitian (conjugate symmetric) arrays.
eigenvalues and eigenvectors of real symmetric or complex Hermitian (conjugate symmetric) arrays.
Similar function in SciPy.
Broadcasting rules apply, see the numpy.linalg documentation for details.
This is implemented using the _geev LAPACK routines which compute the eigenvalues and eigenvectors of general square arrays.
Illustration, using the fact that the eigenvalues of a diagonal matrix are its diagonal elements, that multiplying a matrix on the left by an orthogonal matrix, Q, and on the right by Q.T (the transpose of Q), preserves the eigenvalues of the “middle” matrix. In other words, if Q is orthogonal, then Q * A * Q.T has the same eigenvalues as A :
>>> from numpy import linalg as LA >>> x = np.random.random() >>> Q = np.array([[np.cos(x), -np.sin(x)], [np.sin(x), np.cos(x)]]) >>> LA.norm(Q[0, :]), LA.norm(Q[1, :]), np.dot(Q[0, :],Q[1, :]) (1.0, 1.0, 0.0)
Now multiply a diagonal matrix by Q on one side and by Q.T on the other:
>>> D = np.diag((-1,1)) >>> LA.eigvals(D) array([-1., 1.]) >>> A = np.dot(Q, D) >>> A = np.dot(A, Q.T) >>> LA.eigvals(A) array([ 1., -1.]) # random