# svd¶

tenpy.linalg.np_conserved.svd(a, full_matrices=False, compute_uv=True, cutoff=None, qtotal_LR=[None, None], inner_labels=[None, None], inner_qconj=1)[source]

Singualar value decomposition of an Array a.

Factorizes U, S, VH = svd(a), such that a = U*diag(S)*VH (where * stands for a tensordot() and diag creates an correctly shaped Array with S on the diagonal). For a non-zero cutoff this holds only approximately.

There is a gauge freedom regarding the charges, see also Array.gauge_total_charge(). We ensure contractibility by setting U.legs[1] = VH.legs[0].conj(). Further, we gauge the LegCharge such that U and V have the desired qtotal_LR.

Parameters
• a (Array, shape (M, N)) – The matrix to be decomposed.

• full_matrices (bool) – If False (default), U and V have shapes (M, K) and (K, N), where K=len(S). If True, U and V are full square unitary matrices with shapes (M, M) and (N, N). Note that the arrays are not directly contractible in that case; diag(S) would need to be a rectangluar (M, N) matrix.

• compute_uv (bool) – Whether to compute and return U and V.

• cutoff (None | float) – Keep only singular values which are (strictly) greater than cutoff. (Then the factorization holds only approximately). If None (default), ignored.

• qtotal_LR ([{charges|None}, {charges|None}]) – The desired qtotal for U and VH, respectively. [None, None] (Default) is equivalent to [None, a.qtotal]. A single None entry is replaced the unique charge satisfying the requirement U.qtotal + VH.qtotal = a.qtotal (modulo qmod).

• inner_labels_LR ([{str|None}, {str|None}]) – The first label corresponds to U.legs[1], the second to VH.legs[0].

• inner_qconj ({+1, -1}) – Direction of the charges for the new leg. Default +1. The new LegCharge is constructed such that VH.legs[0].qconj = qconj.

Returns

• U (Array) – Matrix with left singular vectors as columns. Shape (M, M) or (M, K) depending on full_matrices.

• S (1D ndarray) – The singluar values of the array. If no cutoff is given, it has lenght min(M, N).

• VH (Array) – Matrix with right singular vectors as rows. Shape (N, N) or (K, N) depending on full_matrices.