# Kronecker¶

class probnum.linops.Kronecker(A, B)

Bases: LinearOperator

Kronecker product of two linear operators.

The Kronecker product [1] $$A \otimes B$$ of two linear operators $$A$$ and $$B$$ is given by

$\begin{split}A \otimes B = \begin{bmatrix} A_{11} B & \dots & A_{1 m_1} B \\ \vdots & \ddots & \vdots \\ A_{n_11} B & \dots & A_{n_1 m_1} B \end{bmatrix}\end{split}$

where $$A_{ij}v=A(v_j e_i)$$, where $$e_i$$ is the $$i^{\text{th}}$$ unit vector. The result is a new linear operator mapping from $$\mathbb{R}^{n_1n_2}$$ to $$\mathbb{R}^{m_1m_2}$$. By recognizing that $$(A \otimes B)\operatorname{vec}(X) = AXB^{\top}$$, the Kronecker product can be understood as “translation” between matrix multiplication and (row-wise) vectorization.

Parameters:
• A (np.ndarray or LinearOperator) – The first operator.

• B (np.ndarray or LinearOperator) – The second operator.

• dtype (dtype) – Data type of the operator.

References

SymmetricKronecker

The symmetric Kronecker product of two linear operators.

Attributes Summary

 T Transpose of the linear operator. dtype Data type of the linear operator. is_lower_triangular Whether the LinearOperator represents a lower triangular matrix. is_positive_definite Whether the LinearOperator $$L \in \mathbb{R}^{n \times n}$$ is (strictly) positive-definite, i.e. $$x^T L x > 0$$ for $$x \in \mathbb{R}^n$$. is_square Whether input dimension matches output dimension. is_symmetric Whether the LinearOperator $$L$$ is symmetric, i.e. $$L = L^T$$. is_upper_triangular Whether the LinearOperator represents an upper triangular matrix. ndim Number of linear operator dimensions. shape Shape of the linear operator. size Product of the shape entries.

Methods Summary

 __call__(x[, axis]) Apply the linear operator to an input array along a specified axis. astype(dtype[, order, casting, subok, copy]) Cast a linear operator to a different dtype. broadcast_matmat([matmat, method]) Broadcasting for a (implicitly defined) matrix-matrix product. broadcast_matvec([matvec, method]) Broadcasting for a (implicitly defined) matrix-vector product. cholesky([lower]) Computes a Cholesky decomposition of the LinearOperator. cond([p]) Compute the condition number of the linear operator. Determinant of the linear operator. Diagonal of the linear operator. Eigenvalue spectrum of the linear operator. Inverse of the linear operator. Log absolute determinant of the linear operator. Rank of the linear operator. Solves linear systems A @ x = b, where b is either a vector or a (stack of) matrices. Compute or approximate the closest symmetric LinearOperator in the Frobenius norm. todense([cache]) Dense matrix representation of the linear operator. Trace of the linear operator. transpose(*axes) Transpose this linear operator.

Attributes Documentation

T

Transpose of the linear operator.

dtype

Data type of the linear operator.

is_lower_triangular

Whether the LinearOperator represents a lower triangular matrix.

If this is None, it is unknown whether the matrix is lower triangular or not.

is_positive_definite

Whether the LinearOperator $$L \in \mathbb{R}^{n \times n}$$ is (strictly) positive-definite, i.e. $$x^T L x > 0$$ for $$x \in \mathbb{R}^n$$.

If this is None, it is unknown whether the matrix is positive-definite or not. Only symmetric operators can be positive-definite.

Raises:

ValueError – When setting is_positive_definite to True while is_symmetric is False.

is_square

Whether input dimension matches output dimension.

is_symmetric

Whether the LinearOperator $$L$$ is symmetric, i.e. $$L = L^T$$.

If this is None, it is unknown whether the operator is symmetric or not. Only square operators can be symmetric.

Raises:

ValueError – When setting is_symmetric to True on a non-square LinearOperator.

is_upper_triangular

Whether the LinearOperator represents an upper triangular matrix.

If this is None, it is unknown whether the matrix is upper triangular or not.

ndim

Number of linear operator dimensions.

Defined analogously to numpy.ndarray.ndim.

shape

Shape of the linear operator.

Defined as a tuple of the output and input dimension of operator.

size

Product of the shape entries.

Defined analogously to numpy.ndarray.size.

Methods Documentation

__call__(x, axis=None)

Apply the linear operator to an input array along a specified axis.

Parameters:
• x (np.ndarray) – Input array.

• axis (int) – Axis along which to apply the linear operator. Guaranteed to be positive and valid, i.e. axis is a valid index into the shape of x, and x has the correct shape along axis.

Returns:

apply_result – Array resulting in the application of the linear operator to x along axis.

Return type:

np.ndarray

Raises:
• ValueError – If the shape of x is invalid.

• numpy.AxisError – If the axis argument is not within the valid range.

astype(dtype, order='K', casting='unsafe', subok=True, copy=True)

Cast a linear operator to a different dtype.

Parameters:
• dtype (DTypeLike) – Data type to which the linear operator is cast.

• order (str) – Memory layout order of the result.

• casting (str) – Controls what kind of data casting may occur.

• subok (bool) – If True, then sub-classes will be passed-through (default). False is currently not supported for linear operators.

• copy (bool) – Whether to return a new linear operator, even if dtype is the same.

Raises:
Return type:

LinearOperator

Broadcasting for a (implicitly defined) matrix-matrix product.

Convenience function / decorator to broadcast the definition of a matrix-matrix product to stacks of matrices. This can be used to easily construct a new linear operator only from a matrix-matrix product.

Parameters:
• matmat (Callable[[ndarray], ndarray] | None) – Function computing a matrix-matrix product.

• method (bool) – Whether the decorator is being applied to a method or a function.

Return type:

Callable[[ndarray], ndarray] | Callable[[Callable[[ndarray], ndarray]], Callable[[ndarray], ndarray]]

Broadcasting for a (implicitly defined) matrix-vector product.

Convenience function / decorator to broadcast the definition of a matrix-vector product. This can be used to easily construct a new linear operator only from a matrix-vector product.

Parameters:
• matvec (Callable[[ndarray], ndarray] | None) – Function computing a matrix-vector product.

• method (bool) – Whether the decorator is being applied to a method or a function.

Return type:

Callable[[ndarray], ndarray] | Callable[[Callable[[ndarray], ndarray]], Callable[[ndarray], ndarray]]

cholesky(lower=True)

Computes a Cholesky decomposition of the LinearOperator.

The Cholesky decomposition of a symmetric positive-definite matrix $$A \in \mathbb{R}^{n \times n}$$ is given by $$A = L L^T$$, where the unique Cholesky factor $$L \in \mathbb{R}^{n \times n}$$ of $$A$$ is a lower triangular matrix with a positive diagonal.

As a side-effect, this method will set the value of is_positive_definite to True, if the computation of the Cholesky factorization succeeds. Otherwise, is_positive_definite will be set to False.

The result of this computation will be cached. If cholesky() is first called with lower=True first and afterwards with lower=False or vice-versa, the method simply returns the transpose of the cached value.

Parameters:

lower (bool) – If this is set to False, this method computes and returns the upper triangular Cholesky factor $$U = L^T$$, for which $$A = U^T U$$. By default (True), the method computes the lower triangular Cholesky factor $$L$$.

Returns:

The lower or upper Cholesky factor of the LinearOperator, depending on the value of the parameter lower. The result will have its properties is_upper_triangular/is_lower_triangular set accordingly.

Return type:

cholesky_factor

Raises:
cond(p=None)

Compute the condition number of the linear operator.

The condition number of the linear operator with respect to the p norm. It measures how much the solution $$x$$ of the linear system $$Ax=b$$ changes with respect to small changes in $$b$$.

Parameters:

p ({None, 1, , 2, , inf, 'fro'}, optional) –

Order of the norm:

p

norm for matrices

None

2-norm, computed directly via singular value decomposition

’fro’

Frobenius norm

np.inf

max(sum(abs(x), axis=1))

1

max(sum(abs(x), axis=0))

2

2-norm (largest sing. value)

Returns:

The condition number of the linear operator. May be infinite.

Return type:

cond

Raises:

numpy.linalg.LinAlgError – If cond() is called on a non-square matrix.

det()

Determinant of the linear operator.

Returns:

The determinant of the linear operator.

Return type:

det

Raises:

numpy.linalg.LinAlgError – If det() is called on a non-square matrix.

diagonal()

Diagonal of the linear operator.

Return type:

ndarray

eigvals()

Eigenvalue spectrum of the linear operator.

Raises:

numpy.linalg.LinAlgError – If eigvals() is called on a non-square operator.

Return type:

ndarray

inv()

Inverse of the linear operator.

Returns:

inv – Inverse of this linear operator, which is again a LinearOperator.

Return type:

LinearOperator

Raises:

numpy.linalg.LinAlgError – If inv() is called on a non-square linear operator.

logabsdet()

Log absolute determinant of the linear operator.

Returns:

The log absolute determinant of the linear operator.

Return type:

logabsdet

Raises:

numpy.linalg.LinAlgError – If logabsdet() is called on a non-square matrix.

rank()

Rank of the linear operator.

Return type:

int64

solve(b)

Solves linear systems A @ x = b, where b is either a vector or a (stack of) matrices.

This method broadcasts like A.inv() @ b, but it might not produce the exact same result.

Parameters:

b (ArrayLike) – The right-hand side(s) of the linear system(s). This can either be a vector or a (stack of) matrices.

Returns:

The solution(s) of the linear system(s). Depending on the shape of b, x is either a vector or a (stack of) matrices.

Return type:

x

Raises:
• numpy.linalg.LinAlgError – If the linear operator is non-square or not invertible.

• ValueError – If the shape of b does not meet the requirements outlined above.

symmetrize()

Compute or approximate the closest symmetric LinearOperator in the Frobenius norm.

The closest symmetric matrix to a given square matrix $$A$$ in the Frobenius norm is given by

$\operatorname{sym}(A) := \frac{1}{2} (A + A^T).$

However, for efficiency reasons, it is preferrable to approximate this operator in some cases. For example, a Kronecker product $$K = A \otimes B$$ is more efficiently symmetrized by means of

$$\operatorname{sym}(A) \otimes \operatorname{sym}(B) = \operatorname{sym}(K) + \frac{1}{2} \left( \frac{1}{2} \left( A \otimes B^T + A^T \otimes B \right) - \operatorname{sym}(K) \right).$$
Returns:

The closest symmetric LinearOperator in the Frobenius norm, or an approximation, which makes a reasonable trade-off between accuracy and efficiency (see above). The resulting LinearOperator will have its is_symmetric property set to True.

Return type:

symmetrized_linop

Raises:

numpy.linalg.LinAlgError – If this method is called on a non-square LinearOperator.

todense(cache=True)

Dense matrix representation of the linear operator.

This method can be computationally very costly depending on the shape of the linear operator. Use with caution.

Parameters:

cache (bool) – If this is set to True, then the dense matrix representation will be cached and subsequent calls will return the cached value (even if dense is set to False in these subsequent calls).

Returns:

matrix – Matrix representation of the linear operator.

Return type:

np.ndarray

trace()

Trace of the linear operator.

Computes the trace of a square linear operator $$\text{tr}(A) = \sum_{i-1}^n A_{ii}$$.

Returns:

trace – Trace of the linear operator.

Return type:

float

Raises:

numpy.linalg.LinAlgError – If trace() is called on a non-square matrix.

transpose(*axes)

Transpose this linear operator.

Can be abbreviated self.T instead of self.transpose().

Parameters:

*axes (int | Tuple[int]) – Permutation of the axes of the LinearOperator.

Raises:
• ValueError – If the given axis indices do not constitute a valid permutation of the axes.

• numpy.AxisError – If the axis indices are out of bounds.

Return type:

LinearOperator