# Svec¶

class probnum.linops.Svec(dim, check_symmetric=False)

Symmetric vectorization operator.

The column- or row-wise symmetric normalized vectorization operator $$\operatorname{svec}$$ [1] stacking the (normalized) lower/upper triangular components of a symmetric matrix of a linear operator into a vector. It is defined by

$\begin{split}\operatorname{svec}(S) = \begin{bmatrix} S_{11} \\ \sqrt{2} S_{21} \\ \vdots \\ \sqrt{2} S_{n1} \\ S_{22} \\ \sqrt{2} S_{32} \\ \vdots \\ \sqrt{2} S_{n2} \\ \vdots \\ S_{nn} \end{bmatrix}\end{split}$

where $$S$$ is a symmetric linear operator defined on $$\mathbb{R}^n$$.

Parameters: dim (int) – Dimension of the symmetric matrix to be reshaped. check_symmetric (bool, default=False) – Check whether the given matrix or vector corresponds to a symmetric matrix argument. Note, this option can slow down performance.

Notes

It holds that $$Q\operatorname{svec}(S) = \operatorname{vec}(S)$$, where $$Q$$ is a unique matrix with orthonormal rows.

References

 [1] De Klerk, E., Aspects of Semidefinite Programming, Kluwer Academic Publishers, 2002

Attributes Summary

 H Hermitian adjoint. T Transpose this linear operator. ndim

Methods Summary

 __call__(x) Call self as a function. adjoint() Hermitian adjoint. cond([p]) Compute the condition number of the linear operator. det() Determinant of the linear operator. dot(x) Matrix-matrix or matrix-vector multiplication. eigvals() Eigenvalue spectrum of the linear operator. inv() Inverse of the linear operator. logabsdet() Log absolute determinant of the linear operator. matmat(X) Matrix-matrix multiplication. matvec(x) Matrix-vector multiplication. rank() Rank of the linear operator. rmatmat(X) Adjoint matrix-matrix multiplication. rmatvec(x) Adjoint matrix-vector multiplication. todense() Dense matrix representation of the linear operator. trace() Trace of the linear operator. transpose() Transpose this linear operator.

Attributes Documentation

H

Returns the Hermitian adjoint of self, aka the Hermitian conjugate or Hermitian transpose. For a complex matrix, the Hermitian adjoint is equal to the conjugate transpose.

Returns: A_H – Hermitian adjoint of self. LinearOperator
T

Transpose this linear operator.

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

ndim = 2

Methods Documentation

__call__(x)

Call self as a function.

adjoint()

Returns the Hermitian adjoint of self, aka the Hermitian conjugate or Hermitian transpose. For a complex matrix, the Hermitian adjoint is equal to the conjugate transpose.

Returns: A_H – Hermitian adjoint of self. LinearOperator
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:cond – The condition number of the linear operator. May be infinite.
Return type:{float, inf}
det()

Determinant of the linear operator.

dot(x)

Matrix-matrix or matrix-vector multiplication.

Parameters: x (array_like) – 1-d or 2-d array, representing a vector or matrix. Ax – 1-d or 2-d array (depending on the shape of x) that represents the result of applying this linear operator on x. array
eigvals()

Eigenvalue spectrum of the linear operator.

inv()

Inverse of the linear operator.

logabsdet()

Log absolute determinant of the linear operator.

matmat(X)

Matrix-matrix multiplication.

Performs the operation y=A*X where A is an MxN linear operator and X dense N*K matrix or ndarray.

Parameters: X ({matrix, ndarray}) – An array with shape (N,K). Y – A matrix or ndarray with shape (M,K) depending on the type of the X argument. {matrix, ndarray}

Notes

This matmat wraps any user-specified matmat routine or overridden _matmat method to ensure that y has the correct type.

matvec(x)

Matrix-vector multiplication. Performs the operation y=A*x where A is an MxN linear operator and x is a 1-d array or random variable.

Parameters: x ({matrix, ndarray, RandomVariable}) – An array or RandomVariable with shape (N,) or (N,1). y – A matrix or ndarray or RandomVariable with shape (M,) or (M,1) depending on the type and shape of the x argument. {matrix, ndarray}

Notes

This matvec wraps the user-specified matvec routine or overridden _matvec method to ensure that y has the correct shape and type.

rank()

Rank of the linear operator.

rmatmat(X)

Performs the operation y = A^H * x where A is an MxN linear operator and x is a column vector or 1-d array, or 2-d array. The default implementation defers to the adjoint.

Parameters: X ({matrix, ndarray}) – A matrix or 2D array. Y – A matrix or 2D array depending on the type of the input. {matrix, ndarray}

Notes

This rmatmat wraps the user-specified rmatmat routine.

rmatvec(x)

Performs the operation y = A^H * x where A is an MxN linear operator and x is a column vector or 1-d array.

Parameters: x ({matrix, ndarray}) – An array with shape (M,) or (M,1). y – A matrix or ndarray with shape (N,) or (N,1) depending on the type and shape of the x argument. {matrix, ndarray}

Notes

This rmatvec wraps the user-specified rmatvec routine or overridden _rmatvec method to ensure that y has the correct shape and type.

todense()

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.

Returns: matrix – Matrix representation of the linear operator. 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. float ValueError : If trace() is called on a non-square matrix.
transpose()

Transpose this linear operator.

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