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# tf.linalg.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

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.

`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`

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

`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]`

`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 `Tensor`s handled by this `LinearOperator`.
`graph_parents` List of graph dependencies of this `LinearOperator`. (deprecated)

`is_non_singular`

`is_positive_definite`

`is_self_adjoint`

`is_square` Return `True/False` depending on if this operator is square.
`multiplier` The [batch] scalar `Tensor`, `c` in `cI`.
`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`.

`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

### `add_to_tensor`

View source

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`.

### `adjoint`

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 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

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

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

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 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

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

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

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

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 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

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

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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