polynomials_on_simplices.linalg.basis module

Functionality for dealing with bases.

basis_matrix(basis_vectors_array)

Construct the matrix representation of a basis from a list of basis vectors.

Parameters:basis_vectors_array – List of basis vectors. Returns:Matrix representation of basis (square matrix with basis vectors as columns).

basis_vectors(basis)

Get a list of basis vectors from the matrix representation of a basis.

Parameters:basis – Matrix representation of basis (square matrix with basis vectors as columns). Returns:List of basis vectors.

cartesian_basis(dimension=3)

Get the standard Cartesian basis in the specified dimension.

Parameters:dimension (int) – Dimension of Euclidean space. Returns:The identity matrix in \(\mathbb{R}^n\), where n is the desired dimension. Return type:Numpy array

Examples

>>> cartesian_basis(2)
array([[1., 0.],
       [0., 1.]])

cartesian_basis_vector(i, dimension=3)

Get the i:th standard Cartesian basis vector in the specified dimension (3 by default).

Parameters:

• i (int) – Index of the basis vector (in [0, 1, …, dimension - 1]).
• dimension (int) – Dimension of Euclidean space.

Returns:

The i:th Cartesian basis vector.

Return type:

Numpy array

Examples

>>> cartesian_basis_vector(1)
array([0., 1., 0.])

dual_basis(basis)

Compute the dual basis of a given basis, i.e., the basis \(e^i\) such that \(\langle e_i, e^j \rangle = \delta_i^j\).

Parameters:basis – Basis (square matrix with basis vectors as columns). Returns:Dual basis (square matrix with dual basis vectors as columns).

gram_schmidt_orthonormalization(basis, inner_product=<function dot>)

Create an orthonormal basis for an inner product space V from a general basis using the Gram-Schmidt process.

Parameters:

• basis – Input array of basis vectors.
• inner_product – Inner product for the inner product space (a map \(V \times V \to \mathbb{R}\) satisfying the inner product properties).

Returns:

Orthonormal basis (matrix with the basis vectors as columns), where the basis vectors is obtained by Gram-Schmidt orthonormalization of the input basis vectors.

gram_schmidt_orthonormalization_rn(basis)

Create an orthonormal basis from a general basis of \(\mathbb{R}^n\) or an n dimensional subspace of \(\mathbb{R}^m\) using the Gram-Schmidt process.

Parameters:basis (Numpy array) – Input basis (m by n matrix with the basis vectors as columns). Returns:Orthonormal basis (m by n matrix with the basis vectors as columns), where the basis vectors is obtained by Gram-Schmidt orthonormalization of the input basis vectors. Return type:Numpy array

inverse_metric_tensor(basis)

Compute the inverse metric tensor for a basis.

Parameters:basis – Basis (square matrix with basis vectors as columns). Returns:Inverse metric tensor.

metric_tensor(basis)

Compute the metric tensor for a basis.

Parameters:basis – Basis (square matrix with basis vectors as columns). Returns:Metric tensor.

transform(v, basis0, basis1)

Transform a vector or set of vectors from one basis to another.

Parameters:

• v – Representation of the vector(s) in the initial basis (vector or matrix with vectors as columns).
• basis0 – Initial basis to transform from (matrix with basis vectors as columns).
• basis1 – Final basis to transform to (matrix with basis vectors as columns).

Returns:

Representation of the vector(s) in the final basis.

transform_bilinear_form(b, basis0, basis1)

Transform a bilinear form (matrix) from one basis to another.

Parameters:

• b – Representation of the bilinear form in the initial basis (n by n matrix).
• basis0 – Initial basis to transform from (matrix with basis vectors as columns).
• basis1 – Final basis to transform to (matrix with basis vectors as columns).

Returns:

Representation of the bilinear form in the final basis (n by n matrix).

transform_from_basis(v, basis)

Transform a vector or set of vectors in a given coordinate system to Cartesian coordinates.

Parameters:

• v – Vector(s) expressed in the supplied basis (vector or matrix with vectors as columns).
• basis – Basis the vector is expressed in (matrix with basis vectors as columns).

Returns:

Vector(s) in Cartesian coordinates.

transform_to_basis(v, basis)

Transform a vector or set of vectors from Cartesian coordinates to the supplied coordinate system.

Parameters:

• v – Vector(s) expressed in Cartesian coordinates (vector or matrix with vectors as columns).
• basis – Basis to express the vector in (matrix with basis vectors as columns).

Returns:

Vector representation(s) in the supplied basis.