eig(A, *, out=None)¶
Computes the eigenvalue decomposition of a square matrix if it exists.
Letting be or , the eigenvalue decomposition of a square matrix (if it exists) is defined as
This decomposition exists if and only if is diagonalizable. This is the case when all its eigenvalues are different.
Supports input of float, double, cfloat and cdouble dtypes. Also supports batches of matrices, and if
Ais a batch of matrices then the output has the same batch dimensions.
The eigenvalues and eigenvectors of a real matrix may be complex.
When inputs are on a CUDA device, this function synchronizes that device with the CPU.
This function assumes that
Ais diagonalizable (for example, when all the eigenvalues are different). If it is not diagonalizable, the returned eigenvalues will be correct but .
The returned eigenvectors are normalized to have norm 1. Even then, the eigenvectors of a 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 by produces another set of valid eigenvectors of the matrix. For this reason, the loss function shall not depend on the phase of the eigenvectors, as this quantity is not well-defined. This is checked when computing the gradients of this function. As such, when inputs are on a CUDA device, this function synchronizes that device with the CPU when computing the gradients. This is checked when computing the gradients of this function. As such, when inputs are on a CUDA device, the computation of the gradients of this function synchronizes that device with the CPU.
Gradients computed using the eigenvectors tensor will only be finite when
Ahas distinct eigenvalues. Furthermore, if the distance between any two eigenvalues is close to zero, the gradient will be numerically unstable, as it depends on the eigenvalues through the computation of .
torch.linalg.eigh()for a (faster) function that computes the eigenvalue decomposition for Hermitian and symmetric matrices.
torch.linalg.svd()for a function that computes another type of spectral decomposition that works on matrices of any shape.
torch.linalg.qr()for another (much faster) decomposition that works on matrices of any shape.
A (Tensor) – tensor of shape (*, n, n) where * is zero or more batch dimensions consisting of diagonalizable matrices.
- Keyword Arguments
out (tuple, optional) – output tuple of two tensors. Ignored if None. Default: None.
A named tuple (eigenvalues, eigenvectors) which corresponds to and above.
eigenvalues and eigenvectors will always be complex-valued, even when
Ais real. The eigenvectors will be given by the columns of eigenvectors.
>>> A = torch.randn(2, 2, dtype=torch.complex128) >>> A tensor([[ 0.9828+0.3889j, -0.4617+0.3010j], [ 0.1662-0.7435j, -0.6139+0.0562j]], dtype=torch.complex128) >>> L, V = torch.linalg.eig(A) >>> L tensor([ 1.1226+0.5738j, -0.7537-0.1286j], dtype=torch.complex128) >>> V tensor([[ 0.9218+0.0000j, 0.1882-0.2220j], [-0.0270-0.3867j, 0.9567+0.0000j]], dtype=torch.complex128) >>> torch.dist(V @ torch.diag(L) @ torch.linalg.inv(V), A) tensor(7.7119e-16, dtype=torch.float64) >>> A = torch.randn(3, 2, 2, dtype=torch.float64) >>> L, V = torch.linalg.eig(A) >>> torch.dist(V @ torch.diag_embed(L) @ torch.linalg.inv(V), A) tensor(3.2841e-16, dtype=torch.float64)