class_docstr: `LinearOperator` acting like a [batch] square lower triangular matrix.
  
  This operator acts like a [batch] lower triangular matrix `A` with shape
  `[B1,...,Bb, N, N]` for some `b >= 0`.  The first `b` indices index a
  batch member.  For every batch index `(i1,...,ib)`, `A[i1,...,ib, : :]` is
  an `N x N` matrix.
  
  `LinearOperatorLowerTriangular` is initialized with a `Tensor` having
  dimensions `[B1,...,Bb, N, N]`. The upper triangle of the last two
  dimensions is ignored.
  
  ```python
  # Create a 2 x 2 lower-triangular linear operator.
  tril = [[1., 2.], [3., 4.]]
  operator = LinearOperatorLowerTriangular(tril)
  
  # The upper triangle is ignored.
  operator.to_dense()
  ==> [[1., 0.]
       [3., 4.]]
  
  tensor_shape.TensorShape(operator.shape)
  ==> [2, 2]
  
  operator.log_abs_determinant()
  ==> scalar Tensor
  
  x = ... Shape [2, 4] Tensor
  operator.matmul(x)
  ==> Shape [2, 4] Tensor
  
  # Create a [2, 3] batch of 4 x 4 linear operators.
  tril = tf.random.normal(shape=[2, 3, 4, 4])
  operator = LinearOperatorLowerTriangular(tril)
  ```
  
  #### Shape compatibility
  
  This operator acts on [batch] matrix with compatible shape.
  `x` is a batch matrix with compatible shape for `matmul` and `solve` if
  
  ```
  tensor_shape.TensorShape(operator.shape) = [B1,...,Bb] + [N, N],  with b >= 0
  tensor_shape.TensorShape(x.shape) =        [B1,...,Bb] + [N, R],  with R >= 0.
  ```
  
  #### Performance
  
  Suppose `operator` is a `LinearOperatorLowerTriangular` of shape `[N, N]`,
  and `tensor_shape.TensorShape(x.shape) = [N, R]`.  Then
  
  * `operator.matmul(x)` involves `N^2 * R` multiplications.
  * `operator.solve(x)` involves `N * R` size `N` back-substitutions.
  * `operator.determinant()` involves a size `N` `reduce_prod`.
  
  If instead `operator` and `x` have shape `[B1,...,Bb, N, N]` and
  `[B1,...,Bb, N, R]`, every operation increases in complexity by `B1*...*Bb`.
  
  #### Matrix property hints
  
  This `LinearOperator` is initialized with boolean flags of the form `is_X`,
  for `X = non_singular, self_adjoint, positive_definite, square`.
  These have the following meaning:
  
  * If `is_X == True`, callers should expect the operator to have the
    property `X`.  This is a promise that should be fulfilled, but is *not* a
    runtime assert.  For example, finite floating point precision may result
    in these promises being violated.
  * If `is_X == False`, callers should expect the operator to not have `X`.
  * If `is_X == None` (the default), callers should have no expectation either
    way.
docstr: Initialize a `LinearOperatorLowerTriangular`.
  
  Args:
    tril:  Shape `[B1,...,Bb, N, N]` with `b >= 0`, `N >= 0`.
      The lower triangular part of `tril` defines this operator.  The strictly
      upper triangle is ignored.
    is_non_singular:  Expect that this operator is non-singular.
      This operator is non-singular if and only if its diagonal elements are
      all non-zero.
    is_self_adjoint:  Expect that this operator is equal to its hermitian
      transpose.  This operator is self-adjoint only if it is diagonal with
      real-valued diagonal entries.  In this case it is advised to use
      `LinearOperatorDiag`.
    is_positive_definite:  Expect that this operator is positive definite,
      meaning the quadratic form `x^H A x` has positive real part for all
      nonzero `x`.  Note that we do not require the operator to be
      self-adjoint to be positive-definite.  See:
      https://en.wikipedia.org/wiki/Positive-definite_matrix#Extension_for_non-symmetric_matrices
    is_square:  Expect that this operator acts like square [batch] matrices.
    name: A name for this `LinearOperator`.
  
  Raises:
    ValueError:  If `is_square` is `False`.
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code: def __init__(
      self,
      tril,
      is_non_singular=None,
      is_self_adjoint=None,
      is_positive_definite=None,
      is_square=None,
      name="LinearOperatorLowerTriangular",
  ):
      r"""Initialize a `LinearOperatorLowerTriangular`.
  
      Args:
        tril:  Shape `[B1,...,Bb, N, N]` with `b >= 0`, `N >= 0`.
          The lower triangular part of `tril` defines this operator.  The strictly
          upper triangle is ignored.
        is_non_singular:  Expect that this operator is non-singular.
          This operator is non-singular if and only if its diagonal elements are
          all non-zero.
        is_self_adjoint:  Expect that this operator is equal to its hermitian
          transpose.  This operator is self-adjoint only if it is diagonal with
          real-valued diagonal entries.  In this case it is advised to use
          `LinearOperatorDiag`.
        is_positive_definite:  Expect that this operator is positive definite,
          meaning the quadratic form `x^H A x` has positive real part for all
          nonzero `x`.  Note that we do not require the operator to be
          self-adjoint to be positive-definite.  See:
          https://en.wikipedia.org/wiki/Positive-definite_matrix#Extension_for_non-symmetric_matrices
        is_square:  Expect that this operator acts like square [batch] matrices.
        name: A name for this `LinearOperator`.
  
      Raises:
        ValueError:  If `is_square` is `False`.
      """
      parameters = dict(
          tril=tril,
          is_non_singular=is_non_singular,
          is_self_adjoint=is_self_adjoint,
          is_positive_definite=is_positive_definite,
          is_square=is_square,
          name=name,
      )
  
      if is_square is False:
          raise ValueError(
              "Only square lower triangular operators supported at this time."
          )
      is_square = True
  
      with ops.name_scope(name, values=[tril]):
          self._tril = linear_operator_util.convert_nonref_to_tensor(
              tril, name="tril"
          )
          self._check_tril(self._tril)
  
          super(LinearOperatorLowerTriangular, self).__init__(
              dtype=self._tril.dtype,
              is_non_singular=is_non_singular,
              is_self_adjoint=is_self_adjoint,
              is_positive_definite=is_positive_definite,
              is_square=is_square,
              parameters=parameters,
              name=name,
          )
  
imports: from tensorflow_probability.python.internal.backend.numpy import ops
  from tensorflow_probability.python.internal.backend.numpy import numpy_array
  from tensorflow_probability.python.internal.backend.numpy import numpy_math
  from tensorflow_probability.python.internal.backend.numpy import linalg_impl
  from tensorflow_probability.python.internal.backend.numpy.gen import linear_operator
  from tensorflow_probability.python.internal.backend.numpy.gen import linear_operator_util
  from tensorflow_probability.python.internal.backend.numpy.gen import property_hint_util
  import numpy
  from tensorflow_probability.python.internal.backend.numpy import linalg_impl
  from tensorflow_probability.python.internal.backend.numpy import ops
  from tensorflow_probability.python.internal.backend.numpy.gen import tensor_shape
  from tensorflow_probability.python.internal.backend.numpy import private
label_desc: Use image node_id 1 to create a new LinearOperatorLowerTriangular object from inherited base classes: linear_operator with example: obj = LinearOperatorLowerTriangular(tril, is_non_singular, is__adjoint, is_positive_definite, is_square, name)