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# tf.linalg.LinearOperatorScaledIdentity

## Class `LinearOperatorScaledIdentity`

`LinearOperator` acting like a scaled [batch] identity matrix `A = c I`.

This operator acts like a scaled [batch] identity 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 a scaled version of the `N x N` identity matrix.

`LinearOperatorIdentity` is initialized with `num_rows`, and a `multiplier` (a `Tensor`) of shape `[B1,...,Bb]`. `N` is set to `num_rows`, and the `multiplier` determines the scale for each batch member.

``````# Create a 2 x 2 scaled identity matrix.
operator = LinearOperatorIdentity(num_rows=2, multiplier=3.)

operator.to_dense()
==> [[3., 0.]
[0., 3.]]

operator.shape
==> [2, 2]

operator.log_abs_determinant()
==> 2 * Log[3]

x = ... Shape [2, 4] Tensor
operator.matmul(x)
==> 3 * x

y = tf.random_normal(shape=[3, 2, 4])
# Note that y.shape is compatible with operator.shape because operator.shape
# is broadcast to [3, 2, 2].
x = operator.solve(y)
==> 3 * x

# Create a 2-batch of 2x2 identity matrices
operator = LinearOperatorIdentity(num_rows=2, multiplier=5.)
operator.to_dense()
==> [[[5., 0.]
[0., 5.]],
[[5., 0.]
[0., 5.]]]

x = ... Shape [2, 2, 3]
operator.matmul(x)
==> 5 * x

# Here the operator and x have different batch_shape, and are broadcast.
x = ... Shape [1, 2, 3]
operator.matmul(x)
==> 5 * x
``````

### 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]
``````

### Performance

• `operator.matmul(x)` is `O(D1*...*Dd*N*R)`
• `operator.solve(x)` is `O(D1*...*Dd*N*R)`
• `operator.determinant()` is `O(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.

## `__init__`

``````__init__(
num_rows,
multiplier,
is_non_singular=None,
is_positive_definite=None,
is_square=True,
assert_proper_shapes=False,
name='LinearOperatorScaledIdentity'
)
``````

Initialize a `LinearOperatorScaledIdentity`.

The `LinearOperatorScaledIdentity` is initialized with `num_rows`, which determines the size of each identity matrix, and a `multiplier`, which defines `dtype`, batch shape, and scale of each matrix.

#### Args:

• `num_rows`: Scalar non-negative integer `Tensor`. Number of rows in the corresponding identity matrix.
• `multiplier`: `Tensor` of shape `[B1,...,Bb]`, or `[]` (a scalar).
• `is_non_singular`: Expect that this operator is non-singular.
• `is_self_adjoint`: Expect that this operator is equal to its hermitian transpose.
• `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.
• `assert_proper_shapes`: Python `bool`. If `False`, only perform static checks that initialization and method arguments have proper shape. If `True`, and static checks are inconclusive, add asserts to the graph.
• `name`: A name for this `LinearOperator`

#### Raises:

• `ValueError`: If `num_rows` is determined statically to be non-scalar, or negative.

## Properties

### `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.get_shape()[:-2]`

#### Returns:

`TensorShape`, statically determined, may be undefined.

### `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`.

#### Returns:

`Dimension` object.

### `dtype`

The `DType` of `Tensor`s handled by this `LinearOperator`.

### `graph_parents`

List of graph dependencies of this `LinearOperator`.

### `is_square`

Return `True/False` depending on if this operator is square.

### `multiplier`

The [batch] scalar `Tensor`, `c` in `cI`.

### `name`

Name prepended to all ops created by this `LinearOperator`.

### `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`.

#### Returns:

`Dimension` object.

### `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.get_shape()`.

#### Returns:

`TensorShape`, statically determined, may be undefined.

### `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`.

#### Args:

• `name`: A name for this `Op`.

#### Returns:

Python integer, or None if the tensor rank is undefined.

## Methods

### `add_to_tensor`

``````add_to_tensor(
mat,
)
``````

Add matrix represented by this operator to `mat`. Equiv to `I + mat`.

#### Args:

• `mat`: `Tensor` with same `dtype` and shape broadcastable to `self`.
• `name`: A name to give this `Op`.

#### Returns:

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

### `assert_non_singular`

``````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`

``````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`

``````assert_self_adjoint(name='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`

``````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`

``````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`

``````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`

``````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.matrix_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`

``````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`

### `log_abs_determinant`

``````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`

``````matmul(
x,
name='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`

``````matvec(
x,
name='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`

``````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`

``````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`

``````solve(
rhs,
name='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`

``````solvevec(
rhs,
name='solve'
)
``````

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`

``````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`

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

Return a dense (batch) matrix representing this operator.

### `trace`

``````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`.