class_docstr: Class for invariant metrics on Lie groups represented by vectors.
  
  Parameters
  ----------
  space : LieGroup
      Group to equip with the invariant metric
  left : bool
      Whether to use a left or right invariant metric.
      Optional, default: True.
class_docstr_tok: ["Class","for","invariant","metrics","on","Lie","groups","represented","by","vectors",".","Parameters","--","--","--","--","--","space",":","LieGroup","Group","to","equip","with","the","invariant","metric","left",":","bool","Whether","to","use","a","left","or","right","invariant","metric",".","Optional",",","default",":","True","."]
docstr: Compute inner product matrix at the tangent space at a base point.
  
  Parameters
  ----------
  base_point : array-like, shape=[..., dim], optional
      Point in the group (the default is identity).
  
  Returns
  -------
  metric_mat : array-like, shape=[..., dim, dim]
      Metric matrix at base_point.
code: def metric_matrix(self, base_point=None):
      """Compute inner product matrix at the tangent space at a base point.
  
      Parameters
      ----------
      base_point : array-like, shape=[..., dim], optional
          Point in the group (the default is identity).
  
      Returns
      -------
      metric_mat : array-like, shape=[..., dim, dim]
          Metric matrix at base_point.
      """
      if base_point is None:
          return self.metric_mat_at_identity
  
      base_point = self._space.regularize(base_point)
      jacobian = self._space.jacobian_translation(
          point=base_point, left=self.left
      )
  
      inv_jacobian = gs.linalg.inv(jacobian)
      inv_jacobian_transposed = Matrices.transpose(inv_jacobian)
  
      return Matrices.mul(
          inv_jacobian_transposed,
          self.metric_mat_at_identity,
          inv_jacobian,
      )
  
imports: import geomstats.backend
  from geomstats.geometry.matrices import Matrices
  from geomstats.geometry.riemannian_metric import RiemannianMetric
  from geomstats.integrator import integrate
  from geomstats.numerics.geodesic import ExpODESolver, LogShootingSolver
  from geomstats.numerics.ivp import GSIVPIntegrator
  from geomstats.vectorization import repeat_out
label_desc: Use image node_id 3 for calling the _InvariantMetricVector obj's underlying member method code with example usage: obj.metric_matrix(base_point) and returns: Matrices, self