svd

svd(x: array, /, *, full_matrices: bool = True) → Tuple[array, array, array]

Returns a singular value decomposition A = USVh of a matrix (or a stack of matrices) x, where U is a matrix (or a stack of matrices) with orthonormal columns, S is a vector of non-negative numbers (or stack of vectors), and Vh is a matrix (or a stack of matrices) with orthonormal rows.

Parameters:

  • x (array) – input array having shape (..., M, N) and whose innermost two dimensions form matrices on which to perform singular value decomposition. Should have a floating-point data type.

  • full_matrices (bool) – If True, compute full-sized U and Vh, such that U has shape (..., M, M) and Vh has shape (..., N, N). If False, compute on the leading K singular vectors, such that U has shape (..., M, K) and Vh has shape (..., K, N) and where K = min(M, N). Default: True.

Returns:

  • .. – NOTE: once complex numbers are supported, each square matrix must be Hermitian.

  • out (Tuple[array, array, array]) – a namedtuple (U, S, Vh) whose

    • first element must have the field name U and must be an array whose shape depends on the value of full_matrices and contain matrices with orthonormal columns (i.e., the columns are left singular vectors). If full_matrices is True, the array must have shape (..., M, M). If full_matrices is False, the array must have shape (..., M, K), where K = min(M, N). The first x.ndim-2 dimensions must have the same shape as those of the input x.

    • second element must have the field name S and must be an array with shape (..., K) that contains the vector(s) of singular values of length K, where K = min(M, N). For each vector, the singular values must be sorted in descending order by magnitude, such that s[..., 0] is the largest value, s[..., 1] is the second largest value, et cetera. The first x.ndim-2 dimensions must have the same shape as those of the input x.

    • third element must have the field name Vh and must be an array whose shape depends on the value of full_matrices and contain orthonormal rows (i.e., the rows are the right singular vectors and the array is the adjoint). If full_matrices is True, the array must have shape (..., N, N). If full_matrices is False, the array must have shape (..., K, N) where K = min(M, N). The first x.ndim-2 dimensions must have the same shape as those of the input x.

    Each returned array must have the same floating-point data type as x.