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# torch.linalg.matrix_rank¶

torch.linalg.matrix_rank(A, *, atol=None, rtol=None, hermitian=False, out=None)

Computes the numerical rank of a matrix.

The matrix rank is computed as the number of singular values (or eigenvalues in absolute value when hermitian= True) that are greater than $\max(\text{atol}, \sigma_1 * \text{rtol})$ threshold, where $\sigma_1$ is the largest singular value (or eigenvalue).

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.

If hermitian= True, A is assumed to be Hermitian if complex or symmetric if real, but this is not checked internally. Instead, just the lower triangular part of the matrix is used in the computations.

If rtol is not specified and A is a matrix of dimensions (m, n), the relative tolerance is set to be $\text{rtol} = \max(m, n) \varepsilon$ and $\varepsilon$ is the epsilon value for the dtype of A (see finfo). If rtol is not specified and atol is specified to be larger than zero then rtol is set to zero.

If atol or rtol is a torch.Tensor, its shape must be broadcastable to that of the singular values of A as returned by torch.linalg.svdvals().

Note

This function has NumPy compatible variant linalg.matrix_rank(A, tol, hermitian=False). However, use of the positional argument tol is deprecated in favor of atol and rtol.

Note

The matrix rank is computed using a singular value decomposition torch.linalg.svdvals() if hermitian= False (default) and the eigenvalue decomposition torch.linalg.eigvalsh() when hermitian= True. When inputs are on a CUDA device, this function synchronizes that device with the CPU.

Parameters
• A (Tensor) – tensor of shape (*, m, n) where * is zero or more batch dimensions.

• tol (float, Tensor, optional) – [NumPy Compat] Alias for atol. Default: None.

Keyword Arguments
• atol (float, Tensor, optional) – the absolute tolerance value. When None it’s considered to be zero. Default: None.

• rtol (float, Tensor, optional) – the relative tolerance value. See above for the value it takes when None. Default: None.

• hermitian (bool) – indicates whether A is Hermitian if complex or symmetric if real. Default: False.

• out (Tensor, optional) – output tensor. Ignored if None. Default: None.

Examples:

>>> A = torch.eye(10)
>>> torch.linalg.matrix_rank(A)
tensor(10)
>>> B = torch.eye(10)
>>> B[0, 0] = 0
>>> torch.linalg.matrix_rank(B)
tensor(9)

>>> A = torch.randn(4, 3, 2)
>>> torch.linalg.matrix_rank(A)
tensor([2, 2, 2, 2])

>>> A = torch.randn(2, 4, 2, 3)
>>> torch.linalg.matrix_rank(A)
tensor([[2, 2, 2, 2],
[2, 2, 2, 2]])

>>> A = torch.randn(2, 4, 3, 3, dtype=torch.complex64)
>>> torch.linalg.matrix_rank(A)
tensor([[3, 3, 3, 3],
[3, 3, 3, 3]])
>>> torch.linalg.matrix_rank(A, hermitian=True)
tensor([[3, 3, 3, 3],
[3, 3, 3, 3]])
>>> torch.linalg.matrix_rank(A, atol=1.0, rtol=0.0)
tensor([[3, 2, 2, 2],
[1, 2, 1, 2]])
>>> torch.linalg.matrix_rank(A, atol=1.0, rtol=0.0, hermitian=True)
tensor([[2, 2, 2, 1],
[1, 2, 2, 2]])

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