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# torch.linalg.cond

torch.linalg.cond(A, p=None, *, out=None) → Tensor

  Computes the condition number of a matrix with respect to a matrix norm.

  Letting K\mathbb{K}K be R\mathbb{R}R or C\mathbb{C}C, the condition number κ\kappa κ of a matrix A∈K n×n A \in \mathbb{K}^{n \times n}A∈K n×n is defined as

  κ(A)=∥A∥p∥A−1∥p\kappa(A) = \|A\|_p\|A^{-1}\|_p κ(A)=∥A∥p​∥A−1∥p​

  The condition number of `A` measures the numerical stability of the linear system AX = B with respect to a matrix norm.

  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.

  `p` defines the matrix norm that is computed. The following norms are supported:

  `p` | matrix norm
  --- | ---
  None | 2-norm (largest singular value)
  ‘fro’ | Frobenius norm
  ‘nuc’ | nuclear norm
  inf | max(sum(abs(x), dim=1))
  -inf | min(sum(abs(x), dim=1))
  1 | max(sum(abs(x), dim=0))
  -1 | min(sum(abs(x), dim=0))
  2 | largest singular value
  -2 | smallest singular value

  where inf refers to float(‘inf’), NumPy’s inf object, or any equivalent object.

  For `p` is one of (‘fro’, ‘nuc’, inf, -inf, 1, -1), this function uses `torch.linalg.norm()` and `torch.linalg.inv()`. As such, in this case, the matrix (or every matrix in the batch) `A` has to be square and invertible.

  For `p` in (2, -2), this function can be computed in terms of the singular values σ 1≥…≥σ n\sigma_1 \geq \ldots \geq \sigma_n σ 1​≥…≥σ n​

  κ 2(A)=σ 1 σ n κ−2(A)=σ n σ 1\kappa_2(A) = \frac{\sigma_1}{\sigma_n}\qquad \kappa_{-2}(A) = \frac{\sigma_n}{\sigma_1}κ 2​(A)=σ n​σ 1​​κ−2​(A)=σ 1​σ n​​

  In these cases, it is computed using `torch.linalg.svdvals()`. For these norms, the matrix (or every matrix in the batch) `A` may have any shape.

  Note

  When inputs are on a CUDA device, this function synchronizes that device with the CPU if `p` is one of (‘fro’, ‘nuc’, inf, -inf, 1, -1).

  See also

  `torch.linalg.solve()` for a function that solves linear systems of square matrices.

  `torch.linalg.lstsq()` for a function that solves linear systems of general matrices.

  Parameters:

    - A (Tensor) – tensor of shape (*, m, n) where * is zero or more batch dimensions for `p` in (2, -2), and of shape (*, n, n) where every matrix is invertible for `p` in (‘fro’, ‘nuc’, inf, -inf, 1, -1).

    - p (int, inf, -inf, 'fro', 'nuc', optional) – the type of the matrix norm to use in the computations (see above). Default: None

  Keyword Arguments:

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

  Returns:

    A real-valued tensor, even when `A` is complex.

  Raises:

    RuntimeError – if `p` is one of (‘fro’, ‘nuc’, inf, -inf, 1, -1) and the `A` matrix or any matrix in the batch `A` is not square or invertible.

  Examples:

      >>> A = torch.randn(3, 4, 4, dtype=torch.complex64)
      >>> torch.linalg.cond(A)
      >>> A = torch.tensor([[1., 0, -1], [0, 1, 0], [1, 0, 1]])
      >>> torch.linalg.cond(A)
      tensor([1.4142])
      >>> torch.linalg.cond(A, 'fro')
      tensor(3.1623)
      >>> torch.linalg.cond(A, 'nuc')
      tensor(9.2426)
      >>> torch.linalg.cond(A, float('inf'))
      tensor(2.)
      >>> torch.linalg.cond(A, float('-inf'))
      tensor(1.)
      >>> torch.linalg.cond(A, 1)
      tensor(2.)
      >>> torch.linalg.cond(A, -1)
      tensor(1.)
      >>> torch.linalg.cond(A, 2)
      tensor([1.4142])
      >>> torch.linalg.cond(A, -2)
      tensor([0.7071])

      >>> A = torch.randn(2, 3, 3)
      >>> torch.linalg.cond(A)
      tensor([[9.5917],
              [3.2538]])
      >>> A = torch.randn(2, 3, 3, dtype=torch.complex64)
      >>> torch.linalg.cond(A)
      tensor([[4.6245],
              [4.5671]])

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