galgebra.lt

Multivector Linear Transformation

Members

galgebra.lt.Dictionary_to_Matrix(dict_rep, ga)[source]

Convert dictionary representation of linear transformation to matrix

class galgebra.lt.Lt(*args, ga, f=False, mode='g')[source]

A Linear Transformation

Except for the spinor representation the linear transformation is stored as a dictionary with basis vector keys and vector values self.lt_dict so that a is a vector \(a = a^{i}e_{i}\) then

\[\mathtt{self(}a\mathtt{)} = a^{i} * \mathtt{self.lt\_dict[}e_{i}\mathtt{]}.\]

For the spinor representation the linear transformation is stored as the even multivector self.R so that if a is a vector:

self(a) = self.R * a * self.R.rev().
lt_dict

the keys are the basis symbols, \(e_i\), and the dictionary entries are the object vector images (linear combination of sympy non-commutative basis symbols) of the keys so that if L is the linear transformation then:

L(e_i) = self.Ga.mv(L.lt_dict[e_i])
Type:dict
Parameters:
  • ga – Name of metric (geometric algebra)
  • f (bool) – True if Lt if function of coordinates
  • mode (str) – g:general, s:symmetric, a:antisymmetric transformation
galgebra.lt.Matrix_to_dictionary(mat_rep, basis)[source]

Convert matrix representation of linear transformation to dictionary

class galgebra.lt.Mlt(f, Ga, args, fct=False)[source]

A multilinear transformation (mlt) is a multilinear multivector function of a list of vectors (*args) \(F(v_1,...,v_r)\) where for any argument slot \(j\) we have (\(a\) is a scalar and \(u_j\) a vector)

\[\begin{split}F(v_1,...,a*v_j,...,v_r) &= a*F(v_1,...,v_j,...,v_r) \\ F(v_1,...,v_j+u_j,...,v_r) &= F(v_1,...,v_j,...,v_r) + F(v_1,...,u_j,...,v_r).\end{split}\]

If F and G are two Mlts with the same number of argument slots then the sum is

\[(F+G)F(v_1,...,v_r) = F(v_1,...,v_r) + G(v_1,...,v_r).\]

If \(F\) and \(G\) are two Mlts with \(r\) and \(s\) argument slots then their product is

\[(F*G)(v_1,...,v_r,...,v_{r+s}) = F(v_1,...,v_r)*G(v_{r+1},...,v_{r+s}),\]

where \(*\) is any of the multivector multiplicative operations. The derivative of a Mlt with is defined as the directional derivative with respect to the coordinate vector (we assume \(F\) is implicitely a function of the coordinates)

\[F(v_1,...,v_r;v_{r+1}) = (v_{r+1} \bullet \nabla)F(v_1,...,v_j,...,v_r).\]

The contraction of a Mlt between slots \(j\) and \(k\) is defined as the geometric derivative of \(F\) with respect to slot \(k\) and the inner geometric derivative with respect to slot \(j\) (this gives the standard tensor definition of contraction for the case that \(F\) is a scalar function)

\[\begin{split}\operatorname{Contract}(i,j,F) &= \nabla_i \bullet (\nabla_j F(v_1,...,v_i,...,v_j,...,v_r)) \\ &= \nabla_j \bullet (\nabla_i F(v_1,...,v_i,...,v_j,...,v_r)).\end{split}\]

This returns a Mltwith slot \(i\) and \(j\) removed.

Fmt(lcnt=1, title=None)[source]

Set format for printing of Tensors

Parameters:lcnt – Number of components per line

Notes

Usage for tensor T example is:

T.fmt('2','T')

output is:

print 'T = '+str(A)

with two components per line. Works for both standard printing and for latex.