Letting be or , the eigenvalue decomposition of a complex Hermitian or real symmetric matrix is defined as
linalg_eigh(A, UPLO = "L")A list (eigenvalues, eigenvectors) which corresponds to and above.
eigenvalues will always be real-valued, even when A is complex.
It will also be ordered in ascending order.
eigenvectors will have the same dtype as A and will contain the eigenvectors as its columns.
(Tensor): tensor of shape (*, n, n) where * is zero or more batch dimensions
consisting of symmetric or Hermitian matrices.
('L', 'U', optional): controls whether to use the upper or lower triangular part
of A in the computations. Default: 'L'.
The eigenvectors of a symmetric matrix are not unique, nor are they continuous with
respect to A. Due to this lack of uniqueness, different hardware and
software may compute different eigenvectors.
This non-uniqueness is caused by the fact that multiplying an eigenvector by
-1 in the real case or by in the complex
case produces another set of valid eigenvectors of the matrix.
This non-uniqueness problem is even worse when the matrix has repeated eigenvalues.
In this case, one may multiply the associated eigenvectors spanning
the subspace by a rotation matrix and the resulting eigenvectors will be valid
eigenvectors.
Gradients computed using the eigenvectors tensor will only be finite when
A has unique eigenvalues.
Furthermore, if the distance between any two eigvalues is close to zero,
the gradient will be numerically unstable, as it depends on the eigenvalues
through the computation of
.
torch:::math_to_rd(" A = Q \\operatorname{diag}(\\Lambda) Q^{H}\\mathrlap{\\qquad Q \\in \\mathbb{K}^{n \\times n}, \\Lambda \\in \\mathbb{R}^n} ")
where is the conjugate transpose when is complex, and the transpose when is real-valued. is orthogonal in the real case and unitary in the complex case.
Supports input of float, double, cfloat and cdouble dtypes.
Also supports batches of matrices, and if A is a batch of matrices then
the output has the same batch dimensions.
A is assumed to be Hermitian (resp. symmetric), but this is not checked internally, instead:
If UPLO\ = 'L' (default), only the lower triangular part of the matrix is used in the computation.
If UPLO\ = 'U', only the upper triangular part of the matrix is used.
The eigenvalues are returned in ascending order.
linalg_eigvalsh() computes only the eigenvalues values of a Hermitian matrix.
Unlike linalg_eigh(), the gradients of linalg_eigvalsh() are always
numerically stable.
linalg_cholesky() for a different decomposition of a Hermitian matrix.
The Cholesky decomposition gives less information about the matrix but is much faster
to compute than the eigenvalue decomposition.
linalg_eig() for a (slower) function that computes the eigenvalue decomposition
of a not necessarily Hermitian square matrix.
linalg_svd() for a (slower) function that computes the more general SVD
decomposition of matrices of any shape.
linalg_qr() for another (much faster) decomposition that works on general
matrices.
Other linalg:
linalg_cholesky_ex(),
linalg_cholesky(),
linalg_det(),
linalg_eigvalsh(),
linalg_eigvals(),
linalg_eig(),
linalg_householder_product(),
linalg_inv_ex(),
linalg_inv(),
linalg_lstsq(),
linalg_matrix_norm(),
linalg_matrix_power(),
linalg_matrix_rank(),
linalg_multi_dot(),
linalg_norm(),
linalg_pinv(),
linalg_qr(),
linalg_slogdet(),
linalg_solve(),
linalg_svdvals(),
linalg_svd(),
linalg_tensorinv(),
linalg_tensorsolve(),
linalg_vector_norm()
if (torch_is_installed()) {
a <- torch_randn(2, 2)
linalg_eigh(a)
}
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