docstr: The gradient of the similarity constraint function w.r.t. A.
  
  f = \sum_{ij}(x_i-x_j)A(x_i-x_j)' = \sum_{ij}d_ij*A*d_ij'
  df/dA = d(d_ij*A*d_ij')/dA
  
  Note that d_ij*A*d_ij' = tr(d_ij*A*d_ij') = tr(d_ij'*d_ij*A)
  so, d(d_ij*A*d_ij')/dA = d_ij'*d_ij
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code: def _fS1(self, pos_pairs, A):
      r"""The gradient of the similarity constraint function w.r.t. A.
  
      f = \sum_{ij}(x_i-x_j)A(x_i-x_j)' = \sum_{ij}d_ij*A*d_ij'
      df/dA = d(d_ij*A*d_ij')/dA
  
      Note that d_ij*A*d_ij' = tr(d_ij*A*d_ij') = tr(d_ij'*d_ij*A)
      so, d(d_ij*A*d_ij')/dA = d_ij'*d_ij
      """
      diff = pos_pairs[:, 0, :] - pos_pairs[:, 1, :]
      # sum of outer products of all rows in `diff`:
      return np.einsum("ij,ik->jk", diff, diff)
  
imports: import numpy
  from sklearn.base import TransformerMixin
  from sklearn.utils.validation import assert_all_finite
  from .base_metric import _PairsClassifierMixin, MahalanobisMixin
  from .constraints import Constraints, wrap_pairs
  from ._util import components_from_metric, _initialize_metric_mahalanobis
  import warnings
label_desc: Use image node_id 7 for calling the _BaseMMC obj's underlying member method code with example usage: obj._fS1(pos_pairs, A) and returns: np