# tf.linalg.LinearOperatorToeplitz

LinearOperator acting like a [batch] of toeplitz matrices.

Inherits From: LinearOperator

This operator acts like a [batch] Toeplitz 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. This matrix A is not materialized, but for purposes of broadcasting this shape will be relevant.

#### Description in terms of toeplitz matrices

Toeplitz means that A has constant diagonals. Hence, A can be generated with two vectors. One represents the first column of the matrix, and the other represents the first row.

Below is a 4 x 4 example:

A = |a b c d|
|e a b c|
|f e a b|
|g f e a|

#### Example of a Toeplitz operator.

# Create a 3 x 3 Toeplitz operator.
col = [1., 2., 3.]
row = [1., 4., -9.]
operator = LinearOperatorToeplitz(col, row)

operator.to_dense()
==> [[1., 4., -9.],
[2., 1., 4.],
[3., 2., 1.]]

operator.shape
==> [3, 3]

operator.log_abs_determinant()
==> scalar Tensor

x = ... Shape [3, 4] Tensor
operator.matmul(x)
==> Shape [3, 4] Tensor

#### 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 =   [C1,...,Cc] + [N, R],
and [C1,...,Cc] broadcasts with [B1,...,Bb] to [D1,...,Dd]

#### 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.

col Shape [B1,...,Bb, N] Tensor with b >= 0 N >= 0. The first column of the operator. Allowed dtypes: float16, float32, float64, complex64, complex128. Note that the first entry of col is assumed to be the same as the first entry of row.
row Shape [B1,...,Bb, N] Tensor with b >= 0 N >= 0. The first row of the operator. Allowed dtypes: float16, float32, float64, complex64, complex128. Note that the first entry of row is assumed to be the same as the first entry of col.
is_non_singular Expect that this operator is non-singular.
is_self_adjoint Expect that this operator is equal to its hermitian transpose. If diag.dtype is real, this is auto-set to True.
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.

H Returns the adjoint of the current LinearOperator.

Given A representing this LinearOperator, return A*. Note that calling self.adjoint() and self.H are equivalent.

batch_shape TensorShape of batch dimensions of this LinearOperator.

If this operator acts like the batch matrix A with A.shape = [B1,...,Bb, M, N], then this returns TensorShape([B1,...,Bb]), equivalent to A.shape[:-2]

col

domain_dimension Dimension (in the sense of vector spaces) of the domain of this operator.

If this operator acts like the batch matrix A with A.shape = [B1,...,Bb, M, N], then this returns N.

dtype The DType of Tensors handled by this LinearOperator.
graph_parents List of graph dependencies of this LinearOperator. (deprecated)

is_non_singular

is_positive_definite

is_square Return True/False depending on if this operator is square.
range_dimension Dimension (in the sense of vector spaces) of the range of this operator.

If this operator acts like the batch matrix A with A.shape = [B1,...,Bb, M, N], then this returns M.

row

shape TensorShape of this LinearOperator.

If this operator acts like the batch matrix A with A.shape = [B1,...,Bb, M, N], then this returns TensorShape([B1,...,Bb, M, N]), equivalent to A.shape.

tensor_rank Rank (in the sense of tensors) of matrix corresponding to this operator.

If this operator acts like the batch matrix A with A.shape = [B1,...,Bb, M, N], then this returns b + 2.

## Methods

View source

Add matrix represented by this operator to x. Equivalent to A + x.

Args
x Tensor with same dtype and shape broadcastable to self.shape.
name A name to give this Op.

Returns
A Tensor with broadcast shape and same dtype as self.

View source

Returns the adjoint of the current LinearOperator.

Given A representing this LinearOperator, return A*. Note that calling self.adjoint() and self.H are equivalent.

Args
name A name for this Op.

Returns
LinearOperator which represents the adjoint of this LinearOperator.

### assert_non_singular

View source

Returns an Op that asserts this operator is non singular.

This operator is considered non-singular if

ConditionNumber < max{100, range_dimension, domain_dimension} * eps,
eps := np.finfo(self.dtype.as_numpy_dtype).eps

Args
name A string name to prepend to created ops.

Returns
An Assert Op, that, when run, will raise an InvalidArgumentError if the operator is singular.

### assert_positive_definite

View source

Returns an Op that asserts