Computes the eigenvalue decomposition of a square matrix if it exists.
Letting K \mathbb{K} K be R \mathbb{R} R or C \mathbb{C} C, the eigenvalue decomposition of a square matrix A ∈ K n × n A \in \mathbb{K}^{n \times n} A∈Kn×n (if it exists) is defined as
A = V diag ( Λ ) V − 1 V ∈ C n × n , Λ ∈ Cn A = V \operatorname{diag}(\Lambda) V^{-1}\mathrlap{\qquad V \in \mathbb{C}^{n \times n}, \Lambda \in \mathbb{C}^n} A=Vdiag(Λ)V−1V∈Cn×n,Λ∈Cn
This decomposition exists if and only if A A A 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 A is a batch of matrices then the output has the same batch dimensions.
The returned eigenvalues are not guaranteed to be in any specific order.
Note
The eigenvalues and eigenvectors of a real matrix may be complex.
Note
When inputs are on a CUDA device, this function synchronizes that device with the CPU.
Warning
This function assumes that A is diagonalizable (for example, when all the eigenvalues are different). If it is not diagonalizable, the returned eigenvalues will be correct but A ≠ V diag ( Λ ) V − 1 A \neq V \operatorname{diag}(\Lambda)V^{-1} A=Vdiag(Λ)V−1.
Warning
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 e i ϕ , ϕ ∈ R e^{i \phi}, \phi \in \mathbb{R} eiϕ,ϕ∈R 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, the computation of the gradients of this function synchronizes that device with the CPU.
Warning
Gradients computed using the eigenvectors tensor will only be finite when A has 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 λ i \lambda_i λi through the computation of 1 min i ≠ j λ i − λ j \frac{1}{\min_{i \neq j} \lambda_i - \lambda_j} mini=jλi−λj1.
A (Tensor) – tensor of shape (*, n, n) where * is zero or more batch dimensions consisting of diagonalizable matrices.
out (tuple, optional) – output tuple of two tensors. Ignored if None. Default: None.
A named tuple (eigenvalues, eigenvectors) which corresponds to Λ \Lambda Λ and V V V above.
eigenvalues and eigenvectors will always be complex-valued, even when A is real. The eigenvectors will be given by the columns of eigenvectors.
Examples:
>>> 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)
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