# tf.linalg.LinearOperatorToeplitz

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`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.

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<tr>
<td>
`col`
</td>
<td>
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`.
</td>
</tr><tr>
<td>
`row`
</td>
<td>
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`.
</td>
</tr><tr>
<td>
`is_non_singular`
</td>
<td>
Expect that this operator is non-singular.
</td>
</tr><tr>
<td>
</td>
<td>
Expect that this operator is equal to its hermitian
transpose.  If `diag.dtype` is real, this is auto-set to `True`.
</td>
</tr><tr>
<td>
`is_positive_definite`
</td>
<td>
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:
<a href="https://en.wikipedia.org/wiki/Positive-definite_matrix#Extension_for_non-symmetric_matrices">https://en.wikipedia.org/wiki/Positive-definite_matrix#Extension_for_non-symmetric_matrices</a>
</td>
</tr><tr>
<td>
`is_square`
</td>
<td>
Expect that this operator acts like square [batch] matrices.
</td>
</tr><tr>
<td>
`name`
</td>
<td>
A name for this `LinearOperator`.
</td>
</tr>
</table>

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<tr>
<td>
`H`
</td>
<td>
Returns the adjoint of the current `LinearOperator`.

Given `A` representing this `LinearOperator`, return `A*`.
Note that calling `self.adjoint()` and `self.H` are equivalent.
</td>
</tr><tr>
<td>
`batch_shape`
</td>
<td>
`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.get_shape()[:-2]`
</td>
</tr><tr>
<td>
`col`
</td>
<td>

</td>
</tr><tr>
<td>
`domain_dimension`
</td>
<td>
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`.
</td>
</tr><tr>
<td>
`dtype`
</td>
<td>
The `DType` of `Tensor`s handled by this `LinearOperator`.
</td>
</tr><tr>
<td>
`graph_parents`
</td>
<td>
List of graph dependencies of this `LinearOperator`.
</td>
</tr><tr>
<td>
`is_non_singular`
</td>
<td>

</td>
</tr><tr>
<td>
`is_positive_definite`
</td>
<td>

</td>
</tr><tr>
<td>
</td>
<td>

</td>
</tr><tr>
<td>
`is_square`
</td>
<td>
Return `True/False` depending on if this operator is square.
</td>
</tr><tr>
<td>
`range_dimension`
</td>
<td>
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`.
</td>
</tr><tr>
<td>
`row`
</td>
<td>

</td>
</tr><tr>
<td>
`shape`
</td>
<td>
`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.get_shape()`.
</td>
</tr><tr>
<td>
`tensor_rank`
</td>
<td>
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`.
</td>
</tr>
</table>

## Methods

<a target="_blank" href="https://github.com/tensorflow/tensorflow/blob/v2.0.0/tensorflow/python/ops/linalg/linear_operator.py#L1014-L1027">View source</a>

<pre class="devsite-click-to-copy prettyprint lang-py tfo-signature-link">
)
</code></pre>

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

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<tr><th colspan="2">Args</th></tr>

<tr>
<td>
`x`
</td>
<td>
`Tensor` with same `dtype` and shape broadcastable to `self.shape`.
</td>
</tr><tr>
<td>
`name`
</td>
<td>
A name to give this `Op`.
</td>
</tr>
</table>

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<tr><th colspan="2">Returns</th></tr>
<tr class="alt">
<td colspan="2">
A `Tensor` with broadcast shape and same `dtype` as `self`.
</td>
</tr>

</table>

<a target="_blank" href="https://github.com/tensorflow/tensorflow/blob/v2.0.0/tensorflow/python/ops/linalg/linear_operator.py#L870-L885">View source</a>

<pre class="devsite-click-to-copy prettyprint lang-py tfo-signature-link">
)
</code></pre>

Returns the adjoint of the current `LinearOperator`.

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

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<tr><th colspan="2">Args</th></tr>

<tr>
<td>
`name`
</td>
<td>
A name for this `Op`.
</td>
</tr>
</table>

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<tr><th colspan="2">Returns</th></tr>
<tr class="alt">
<td colspan="2">
`LinearOperator` which represents the adjoint of this `LinearOperator`.
</td>
</tr>

</table>

<h3 id="assert_non_singular"><code>assert_non_singular</code></h3>

<a target="_blank" href="https://github.com/tensorflow/tensorflow/blob/v2.0.0/tensorflow/python/ops/linalg/linear_operator.py#L484-L502">View source</a>

<pre class="devsite-click-to-copy prettyprint lang-py tfo-signature-link">
<code>assert_non_singular(
name='assert_non_singular'
)
</code></pre>

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

``````
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<tr>
<td>
`name`
</td>
<td>
A string name to prepend to created ops.
</td>
</tr>
</table>

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<tr><th colspan="2">Returns</th></tr>
<tr class="alt">
<td colspan="2">
An `Assert` `Op`, that, when run, will raise an `InvalidArgumentError` if
the operator is singular.
</td>
</tr>

</table>

<h3 id="assert_positive_definite"><code>assert_positive_definite</code></h3>

<a target="_blank" href="https://github.com/tensorflow/tensorflow/blob/v2.0.0/tensorflow/python/ops/linalg/linear_operator.py#L520-L535">View source</a>

<pre class="devsite-click-to-copy prettyprint lang-py tfo-signature-link">
<code>assert_positive_definite(
name='assert_positive_definite'
)
</code></pre>

Returns an `Op` that asserts this operator is positive definite.

Here, positive definite means that 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.

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<tr><th colspan="2">Args</th></tr>

<tr>
<td>
`name`
</td>
<td>
A name to give this `Op`.
</td>
</tr>
</table>

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<colgroup><col width="214px"><col></colgroup>
<tr><th colspan="2">Returns</th></tr>
<tr class="alt">
<td colspan="2">
An `Assert` `Op`, that, when run, will raise an `InvalidArgumentError` if
the operator is not positive definite.
</td>
</tr>

</table>

<a target="_blank" href="https://github.com/tensorflow/tensorflow/blob/v2.0.0/tensorflow/python/ops/linalg/linear_operator.py#L547-L561">View source</a>

<pre class="devsite-click-to-copy prettyprint lang-py tfo-signature-link">
)
</code></pre>

Returns an `Op` that asserts this operator is self-adjoint.

Here we check that this operator is *exactly* equal to its hermitian
transpose.

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<tr><th colspan="2">Args</th></tr>

<tr>
<td>
`name`
</td>
<td>
A string name to prepend to created ops.
</td>
</tr>
</table>

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<colgroup><col width="214px"><col></colgroup>
<tr><th colspan="2">Returns</th></tr>
<tr class="alt">
<td colspan="2">
An `Assert` `Op`, that, when run, will raise an `InvalidArgumentError` if
the operator is not self-adjoint.
</td>
</tr>

</table>

<h3 id="batch_shape_tensor"><code>batch_shape_tensor</code></h3>

<a target="_blank" href="https://github.com/tensorflow/tensorflow/blob/v2.0.0/tensorflow/python/ops/linalg/linear_operator.py#L318-L338">View source</a>

<pre class="devsite-click-to-copy prettyprint lang-py tfo-signature-link">
<code>batch_shape_tensor(
name='batch_shape_tensor'
)
</code></pre>

Shape of batch dimensions of this operator, determined at runtime.

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

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<tr>
<td>
`name`
</td>
<td>
A name for this `Op`.
</td>
</tr>
</table>

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<tr><th colspan="2">Returns</th></tr>
<tr class="alt">
<td colspan="2">
`int32` `Tensor`
</td>
</tr>

</table>

<h3 id="cholesky"><code>cholesky</code></h3>

<a target="_blank" href="https://github.com/tensorflow/tensorflow/blob/v2.0.0/tensorflow/python/ops/linalg/linear_operator.py#L915-L938">View source</a>

<pre class="devsite-click-to-copy prettyprint lang-py tfo-signature-link">
<code>cholesky(
name='cholesky'
)
</code></pre>

Returns a Cholesky factor as a `LinearOperator`.

Given `A` representing this `LinearOperator`, if `A` is positive definite
self-adjoint, return `L`, where `A = L L^T`, i.e. the cholesky
decomposition.

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<tr><th colspan="2">Args</th></tr>

<tr>
<td>
`name`
</td>
<td>
A name for this `Op`.
</td>
</tr>
</table>

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<tr><th colspan="2">Returns</th></tr>
<tr class="alt">
<td colspan="2">
`LinearOperator` which represents the lower triangular matrix
in the Cholesky decomposition.
</td>
</tr>

</table>

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<tr><th colspan="2">Raises</th></tr>

<tr>
<td>
`ValueError`
</td>
<td>
When the `LinearOperator` is not hinted to be positive
definite and self adjoint.
</td>
</tr>
</table>

<h3 id="determinant"><code>determinant</code></h3>

<a target="_blank" href="https://github.com/tensorflow/tensorflow/blob/v2.0.0/tensorflow/python/ops/linalg/linear_operator.py#L677-L694">View source</a>

<pre class="devsite-click-to-copy prettyprint lang-py tfo-signature-link">
<code>determinant(
name='det'
)
</code></pre>

Determinant for every batch member.

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<tr>
<td>
`name`
</td>
<td>
A name for this `Op`.
</td>
</tr>
</table>

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<tr><th colspan="2">Returns</th></tr>
<tr class="alt">
<td colspan="2">
`Tensor` with shape `self.batch_shape` and same `dtype` as `self`.
</td>
</tr>

</table>

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<tr><th colspan="2">Raises</th></tr>

<tr>
<td>
`NotImplementedError`
</td>
<td>
If `self.is_square` is `False`.
</td>
</tr>
</table>

<h3 id="diag_part"><code>diag_part</code></h3>

<a target="_blank" href="https://github.com/tensorflow/tensorflow/blob/v2.0.0/tensorflow/python/ops/linalg/linear_operator.py#L965-L991">View source</a>

<pre class="devsite-click-to-copy prettyprint lang-py tfo-signature-link">
<code>diag_part(
name='diag_part'
)
</code></pre>

Efficiently get the [batch] diagonal part of this operator.

If this operator has shape `[B1,...,Bb, M, N]`, this returns a
`Tensor` `diagonal`, of shape `[B1,...,Bb, min(M, N)]`, where
`diagonal[b1,...,bb, i] = self.to_dense()[b1,...,bb, i, i]`.

``````

my_operator = LinearOperatorDiag([1., 2.])

# Efficiently get the diagonal

my_operator.diag_part() ==> [1., 2.]

# Equivalent, but inefficient method

tf.linalg.diag_part(my_operator.to_dense()) ==> [1., 2.]

``````
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<tr><th colspan="2">Args</th></tr>

<tr>
<td>
`name`
</td>
<td>
A name for this `Op`.
</td>
</tr>
</table>

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<tr><th colspan="2">Returns</th></tr>

<tr>
<td>
`diag_part`
</td>
<td>
A `Tensor` of same `dtype` as self.
</td>
</tr>
</table>

<h3 id="domain_dimension_tensor"><code>domain_dimension_tensor</code></h3>

<a target="_blank" href="https://github.com/tensorflow/tensorflow/blob/v2.0.0/tensorflow/python/ops/linalg/linear_operator.py#L393-L414">View source</a>

<pre class="devsite-click-to-copy prettyprint lang-py tfo-signature-link">
<code>domain_dimension_tensor(
name='domain_dimension_tensor'
)
</code></pre>

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

Determined at runtime.

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

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<tr><th colspan="2">Args</th></tr>

<tr>
<td>
`name`
</td>
<td>
A name for this `Op`.
</td>
</tr>
</table>

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<tr><th colspan="2">Returns</th></tr>
<tr class="alt">
<td colspan="2">
`int32` `Tensor`
</td>
</tr>

</table>

<h3 id="inverse"><code>inverse</code></h3>

<a target="_blank" href="https://github.com/tensorflow/tensorflow/blob/v2.0.0/tensorflow/python/ops/linalg/linear_operator.py#L890-L913">View source</a>

<pre class="devsite-click-to-copy prettyprint lang-py tfo-signature-link">
<code>inverse(
name='inverse'
)
</code></pre>

Returns the Inverse of this `LinearOperator`.

Given `A` representing this `LinearOperator`, return a `LinearOperator`
representing `A^-1`.

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<tr>
<td>
`name`
</td>
<td>
A name scope to use for ops added by this method.
</td>
</tr>
</table>

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<tr><th colspan="2">Returns</th></tr>
<tr class="alt">
<td colspan="2">
`LinearOperator` representing inverse of this matrix.
</td>
</tr>

</table>

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<tr><th colspan="2">Raises</th></tr>

<tr>
<td>
`ValueError`
</td>
<td>
When the `LinearOperator` is not hinted to be `non_singular`.
</td>
</tr>
</table>

<h3 id="log_abs_determinant"><code>log_abs_determinant</code></h3>

<a target="_blank" href="https://github.com/tensorflow/tensorflow/blob/v2.0.0/tensorflow/python/ops/linalg/linear_operator.py#L706-L723">View source</a>

<pre class="devsite-click-to-copy prettyprint lang-py tfo-signature-link">
<code>log_abs_determinant(
name='log_abs_det'
)
</code></pre>

Log absolute value of determinant for every batch member.

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<tr><th colspan="2">Args</th></tr>

<tr>
<td>
`name`
</td>
<td>
A name for this `Op`.
</td>
</tr>
</table>

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<colgroup><col width="214px"><col></colgroup>
<tr><th colspan="2">Returns</th></tr>
<tr class="alt">
<td colspan="2">
`Tensor` with shape `self.batch_shape` and same `dtype` as `self`.
</td>
</tr>

</table>

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<colgroup><col width="214px"><col></colgroup>
<tr><th colspan="2">Raises</th></tr>

<tr>
<td>
`NotImplementedError`
</td>
<td>
If `self.is_square` is `False`.
</td>
</tr>
</table>

<h3 id="matmul"><code>matmul</code></h3>

<a target="_blank" href="https://github.com/tensorflow/tensorflow/blob/v2.0.0/tensorflow/python/ops/linalg/linear_operator.py#L574-L627">View source</a>

<pre class="devsite-click-to-copy prettyprint lang-py tfo-signature-link">
<code>matmul(
)
</code></pre>

Transform [batch] matrix `x` with left multiplication:  `x --> Ax`.

```python
# Make an operator acting like batch matrix A.  Assume A.shape = [..., M, N]
operator = LinearOperator(...)
operator.shape = [..., M, N]

X = ... # shape [..., N, R], batch matrix, R > 0.

Y = operator.matmul(X)
Y.shape
==> [..., M, R]

Y[..., :, r] = sum_j A[..., :, j] X[j, r]
``````

Args
`x` `LinearOperator` or `Tensor` with compatible shape and same `dtype` as `self`. See class docstring for definition of compatibility.
`adjoint` Python `bool`. If `True`, left multiply by the adjoint: `A^H x`.
`adjoint_arg` Python `bool`. If `True`, compute `A x^H` where `x^H` is the hermitian transpose (transposition and complex conjugation).
`name` A name for this `Op`.

Returns
A `LinearOperator` or `Tensor` with shape `[..., M, R]` and same `dtype` as `self`.

### `matvec`

View source

Transform [batch] vector `x` with left multiplication: `x --> Ax`.

``````# Make an operator acting like batch matric A.  Assume A.shape = [..., M, N]
operator = LinearOperator(...)

X = ... # shape [..., N], batch vector

Y = operator.matvec(X)
Y.shape
==> [..., M]

Y[..., :] = sum_j A[..., :, j] X[..., j]
``````

Args
`x` `Tensor` with compatible shape and same `dtype` as `self`. `x` is treated as a [batch] vector meaning for every set of leading dimensions, the last dimension defines a vector. See class docstring for definition of compatibility.
`adjoint` Python `bool`. If `True`, left multiply by the adjoint: `A^H x`.
`name` A name for this `Op`.

Returns
A `Tensor` with shape `[..., M]` and same `dtype` as `self`.

### `range_dimension_tensor`

View source

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

Determined at runtime.

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

Args
`name` A name for this `Op`.

Returns
`int32` `Tensor`

### `shape_tensor`

View source

Shape of this `LinearOperator`, determined at runtime.

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

Args
`name` A name for this `Op`.

Returns
`int32` `Tensor`

### `solve`

View source

Solve (exact or approx) `R` (batch) systems of equations: `A X = rhs`.

The returned `Tensor` will be close to an exact solution if `A` is well conditioned. Otherwise closeness will vary. See class docstring for details.

#### Examples:

``````# Make an operator acting like batch matrix A.  Assume A.shape = [..., M, N]
operator = LinearOperator(...)
operator.shape = [..., M, N]

# Solve R > 0 linear systems for every member of the batch.
RHS = ... # shape [..., M, R]

X = operator.solve(RHS)
# X[..., :, r] is the solution to the r'th linear system
# sum_j A[..., :, j] X[..., j, r] = RHS[..., :, r]

operator.matmul(X)
==> RHS
``````

Args
`rhs` `Tensor` with same `dtype` as this operator and compatible shape. `rhs` is treated like a [batch] matrix meaning for every set of leading dimensions, the last two dimensions defines a matrix. See class docstring for definition of compatibility.
`adjoint` Python `bool`. If `True`, solve the system involving the adjoint of this `LinearOperator`: `A^H X = rhs`.
`adjoint_arg` Python `bool`. If `True`, solve `A X = rhs^H` where `rhs^H` is the hermitian transpose (transposition and complex conjugation).
`name` A name scope to use for ops added by this method.

Returns
`Tensor` with shape `[...,N, R]` and same `dtype` as `rhs`.

Raises
`NotImplementedError` If `self.is_non_singular` or `is_square` is False.

### `solvevec`

View source

Solve single equation with best effort: `A X = rhs`.

The returned `Tensor` will be close to an exact solution if `A` is well conditioned. Otherwise closeness will vary. See class docstring for details.

#### Examples:

``````# Make an operator acting like batch matrix A.  Assume A.shape = [..., M, N]
operator = LinearOperator(...)
operator.shape = [..., M, N]

# Solve one linear system for every member of the batch.
RHS = ... # shape [..., M]

X = operator.solvevec(RHS)
# X is the solution to the linear system
# sum_j A[..., :, j] X[..., j] = RHS[..., :]

operator.matvec(X)
==> RHS
``````

Args
`rhs` `Tensor` with same `dtype` as this operator. `rhs` is treated like a [batch] vector meaning for every set of leading dimensions, the last dimension defines a vector. See class docstring for definition of compatibility regarding batch dimensions.
`adjoint` Python `bool`. If `True`, solve the system involving the adjoint of this `LinearOperator`: `A^H X = rhs`.
`name` A name scope to use for ops added by this method.

Returns
`Tensor` with shape `[...,N]` and same `dtype` as `rhs`.

Raises
`NotImplementedError` If `self.is_non_singular` or `is_square` is False.

### `tensor_rank_tensor`

View source

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`.

Args
`name` A name for this `Op`.

Returns
`int32` `Tensor`, determined at runtime.

### `to_dense`

View source

Return a dense (batch) matrix representing this operator.

### `trace`

View source

Trace of the linear operator, equal to sum of `self.diag_part()`.

If the operator is square, this is also the sum of the eigenvalues.

Args
`name` A name for this `Op`.

Returns
Shape `[B1,...,Bb]` `Tensor` of same `dtype` as `self`.

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[{ "type": "thumb-up", "id": "easyToUnderstand", "label":"Easy to understand" },{ "type": "thumb-up", "id": "solvedMyProblem", "label":"Solved my problem" },{ "type": "thumb-up", "id": "otherUp", "label":"Other" }]