# LostTech.TensorFlow : API Documentation

Type LinearOperatorLowerTriangular

Namespace tensorflow.linalg

Parent LinearOperator

Interfaces ILinearOperatorLowerTriangular

`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. #### Shape compatibility

This operator acts on [batch] matrix with compatible shape. `x` is a batch matrix with compatible shape for `matmul` and `solve` if

``` operator.shape = [B1,...,Bb] + [N, N], with b >= 0 x.shape = [B1,...,Bb] + [N, R], with R >= 0. ```

#### Performance

Suppose `operator` is a `LinearOperatorLowerTriangular` of shape `[N, N]`, and `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.
Show Example
```# 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.]]  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) ```