Missed TensorFlow Dev Summit? Check out the video playlist.

# tf.linalg.LinearOperatorPermutation

`LinearOperator` acting like a [batch] of permutation matrices.

Inherits From: `LinearOperator`

``````tf.linalg.LinearOperatorPermutation(
is_positive_definite=None, is_square=None, name='LinearOperatorPermutation'
)
``````

This operator acts like a [batch] of permutations 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.

`LinearOperatorPermutation` is initialized with a (batch) vector.

A permutation, is defined by an integer vector `v` whose values are unique and are in the range `[0, ... n]`. Applying the permutation on an input matrix has the folllowing meaning: the value of `v` at index `i` says to move the `v[i]`-th row of the input matrix to the `i`-th row. Because all values are unique, this will result in a permutation of the rows the input matrix. Note, that the permutation vector `v` has the same semantics as `tf.transpose`.

``````# Create a 3 x 3 permutation matrix that swaps the last two columns.
vec = [0, 2, 1]
operator = LinearOperatorPermutation(vec)

operator.to_dense()
==> [[1., 0., 0.]
[0., 0., 1.]
[0., 1., 0.]]

operator.shape
==> [3, 3]

# This will be zero.
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.

#### Args:

* <b>`perm`</b>:  Shape `[B1,...,Bb, N]` Integer `Tensor` with `b >= 0`
`N >= 0`. An integer vector that represents the permutation to apply.
Note that this argument is same as <a href="../../tf/transpose"><code>tf.transpose</code></a>. However, this
permutation is applied on the rows, while the permutation in
<a href="../../tf/transpose"><code>tf.transpose</code></a> is applied on the dimensions of the `Tensor`. `perm`
is required to have unique entries from `{0, 1, ... N-1}`.
* <b>`dtype`</b>: The `dtype` of arguments to this operator. Default: `float32`.
Allowed dtypes: `float16`, `float32`, `float64`, `complex64`,
`complex128`.
* <b>`is_non_singular`</b>:  Expect that this operator is non-singular.
* <b>`is_self_adjoint`</b>:  Expect that this operator is equal to its hermitian
transpose.  This is autoset to true
* <b>`is_positive_definite`</b>:  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
https://en.wikipedia.org/wiki/Positive-definite_matrix#Extension_for_non-symmetric_matrices
This is autoset to false.
* <b>`is_square`</b>:  Expect that this operator acts like square [batch] matrices.
This is autoset to true.
* <b>`name`</b>: A name for this `LinearOperator`.

#### Attributes:

* <b>`H`</b>:   Returns the adjoint of the current `LinearOperator`.

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

* <b>`batch_shape`</b>:   `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]`

* <b>`domain_dimension`</b>:   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`.

* <b>`dtype`</b>:   The `DType` of `Tensor`s handled by this `LinearOperator`.
* <b>`graph_parents`</b>:   List of graph dependencies of this `LinearOperator`. (deprecated)

Warning: THIS FUNCTION IS DEPRECATED. It will be removed in a future version.
Instructions for updating:
Do not call `graph_parents`.

* <b>`is_non_singular`</b>
* <b>`is_positive_definite`</b>
* <b>`is_square`</b>:   Return `True/False` depending on if this operator is square.
* <b>`perm`</b>
* <b>`range_dimension`</b>:   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`.

* <b>`shape`</b>:   `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`.

* <b>`tensor_rank`</b>:   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`.

#### Raises:

* <b>`ValueError`</b>:  `is_self_adjoint` is not `True`, `is_positive_definite` is
not `False` or `is_square` is not `True`.

## Methods

<a target="_blank" href="https://github.com/tensorflow/tensorflow/blob/v2.1.0/tensorflow/python/ops/linalg/linear_operator.py#L1039-L1052">View source</a>

```python
)
``````

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`.

### `adjoint`

View source

``````adjoint(
)
``````

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

``````assert_non_singular(
name='assert_non_singular'
)
``````

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

``````assert_positive_definite(
name='assert_positive_definite'
)
``````

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.

#### Args:

• `name`: A name to give this `Op`.

#### Returns:

An `Assert` `Op`, that, when run, will raise an `InvalidArgumentError` if the operator is not positive definite.

### `assert_self_adjoint`

View source

``````assert_self_adjoint(
)
``````

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

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

#### 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 not self-adjoint.

### `batch_shape_tensor`

View source

``````batch_shape_tensor(
name='batch_shape_tensor'
)
``````

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]`.

#### Args:

• `name`: A name for this `Op`.

#### Returns:

`int32` `Tensor`

### `cholesky`

View source

``````cholesky(
name='cholesky'
)
``````

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.

#### Args:

• `name`: A name for this `Op`.

#### Returns:

`LinearOperator` which represents the lower triangular matrix in the Cholesky decomposition.

#### Raises:

• `ValueError`: When the `LinearOperator` is not hinted to be positive definite and self adjoint.

### `determinant`

View source

``````determinant(
name='det'
)
``````

Determinant for every batch member.

#### Args:

• `name`: A name for this `Op`.

#### Returns:

`Tensor` with shape `self.batch_shape` and same `dtype` as `self`.

#### Raises:

• `NotImplementedError`: If `self.is_square` is `False`.

### `diag_part`

View source

``````diag_part(
name='diag_part'
)
``````

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.]
``````

#### Args:

• `name`: A name for this `Op`.

#### Returns:

• `diag_part`: A `Tensor` of same `dtype` as self.

### `domain_dimension_tensor`

View source

``````domain_dimension_tensor(
name='domain_dimension_tensor'
)
``````

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`.

#### Args:

• `name`: A name for this `Op`.

#### Returns:

`int32` `Tensor`

### `eigvals`

View source

``````eigvals(
name='eigvals'
)
``````

Returns the eigenvalues of this linear operator.

If the operator is marked as self-adjoint (via `is_self_adjoint`) this computation can be more efficient.

#### Args:

• `name`: A name for this `Op`.

#### Returns:

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

### `inverse`

View source

``````inverse(
name='inverse'
)
``````

Returns the Inverse of this `LinearOperator`.

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

#### Args:

• `name`: A name scope to use for ops added by this method.

#### Returns:

`LinearOperator` representing inverse of this matrix.

#### Raises:

• `ValueError`: When the `LinearOperator` is not hinted to be `non_singular`.

### `log_abs_determinant`

View source

``````log_abs_determinant(
name='log_abs_det'
)
``````

Log absolute value of determinant for every batch member.

#### Args:

• `name`: A name for this `Op`.

#### Returns:

`Tensor` with shape `self.batch_shape` and same `dtype` as `self`.

#### Raises:

• `NotImplementedError`: If `self.is_square` is `False`.

### `matmul`

View source

``````matmul(
)
``````

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

``````# 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

``````matvec(
)
``````

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

``````range_dimension_tensor(
name='range_dimension_tensor'
)
``````

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_tensor(
name='shape_tensor'
)
``````

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(
)
``````

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

``````solvevec(
)
``````

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

``````tensor_rank_tensor(
name='tensor_rank_tensor'
)
``````

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

``````to_dense(
name='to_dense'
)
``````

Return a dense (batch) matrix representing this operator.

### `trace`

View source

``````trace(
name='trace'
)
``````

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`.