# torch.symeig¶

torch.symeig(input, eigenvectors=False, upper=True, *, out=None) -> (Tensor, Tensor)

This function returns eigenvalues and eigenvectors of a real symmetric matrix input or a batch of real symmetric matrices, represented by a namedtuple (eigenvalues, eigenvectors).

This function calculates all eigenvalues (and vectors) of input such that $\text{input} = V \text{diag}(e) V^T$ .

The boolean argument eigenvectors defines computation of both eigenvectors and eigenvalues or eigenvalues only.

If it is False, only eigenvalues are computed. If it is True, both eigenvalues and eigenvectors are computed.

Since the input matrix input is supposed to be symmetric, only the upper triangular portion is used by default.

If upper is False, then lower triangular portion is used.

Note

The eigenvalues are returned in ascending order. If input is a batch of matrices, then the eigenvalues of each matrix in the batch is returned in ascending order.

Note

Irrespective of the original strides, the returned matrix V will be transposed, i.e. with strides V.contiguous().transpose(-1, -2).stride().

Note

Extra care needs to be taken when backward through outputs. Such operation is really only stable when all eigenvalues are distinct. Otherwise, NaN can appear as the gradients are not properly defined.

Parameters
• input (Tensor) – the input tensor of size $(*, n, n)$ where * is zero or more batch dimensions consisting of symmetric matrices.

• eigenvectors (boolean, optional) – controls whether eigenvectors have to be computed

• upper (boolean, optional) – controls whether to consider upper-triangular or lower-triangular region

Keyword Arguments

out (tuple, optional) – the output tuple of (Tensor, Tensor)

Returns

A namedtuple (eigenvalues, eigenvectors) containing

• eigenvalues (Tensor): Shape $(*, m)$ . The eigenvalues in ascending order.

• eigenvectors (Tensor): Shape $(*, m, m)$ . If eigenvectors=False, it’s an empty tensor. Otherwise, this tensor contains the orthonormal eigenvectors of the input.

Return type

(Tensor, Tensor)

Examples:

>>> a = torch.randn(5, 5)
>>> a = a + a.t()  # To make a symmetric
>>> a
tensor([[-5.7827,  4.4559, -0.2344, -1.7123, -1.8330],
[ 4.4559,  1.4250, -2.8636, -3.2100, -0.1798],
[-0.2344, -2.8636,  1.7112, -5.5785,  7.1988],
[-1.7123, -3.2100, -5.5785, -2.6227,  3.1036],
[-1.8330, -0.1798,  7.1988,  3.1036, -5.1453]])
>>> e, v = torch.symeig(a, eigenvectors=True)
>>> e
tensor([-13.7012,  -7.7497,  -2.3163,   5.2477,   8.1050])
>>> v
tensor([[ 0.1643,  0.9034, -0.0291,  0.3508,  0.1817],
[-0.2417, -0.3071, -0.5081,  0.6534,  0.4026],
[-0.5176,  0.1223, -0.0220,  0.3295, -0.7798],
[-0.4850,  0.2695, -0.5773, -0.5840,  0.1337],
[ 0.6415, -0.0447, -0.6381, -0.0193, -0.4230]])
>>> a_big = torch.randn(5, 2, 2)
>>> a_big = a_big + a_big.transpose(-2, -1)  # To make a_big symmetric
>>> e, v = a_big.symeig(eigenvectors=True)
>>> torch.allclose(torch.matmul(v, torch.matmul(e.diag_embed(), v.transpose(-2, -1))), a_big)
True