# Matrix multiplication with a vector in EuclideanSpace

Hello

Oversimplified version of my question is "How can I multiply a vector with a matrix?"

E.<x,y,z>=EuclideanSpace(start_index=0)
vf=E.vector_field([x*y,x^2,z**2]) # the components are for testing purposes

M=Matrix(RR,3,3);M[:]=1 # this is just a test
# Neither this
M*vf
# nor this works
M*vf.at(E((3,2,1)))


The only work around I could think of is

v=vf.at(E((3,2,1)))
q=vector([v[0],v[1],v[2]])
M*q


or directly on the vector field

v=vector([v[0].expr(), v[1].expr(), v[2].expr()])
M*q


With this approach I have to put the transformed components back in the vector field defined within EuclideanSpace.

I would like to use EuclideanSpace to work with vectors.

Is there an easy way to handle this? Basically I am looking for an easy way to transform a vector field defined in EuclideanSpace?

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You have to convert the matrix M to an endomorphism of the Euclidean space and apply the latter to the vector field. Given the canonical identification between endomorphisms and tensors of type (1,1), the conversion is performed as follows:

A = E.tensor_field(1, 1, M)


Here is the full example:

sage: E.<x,y,z> = EuclideanSpace(start_index=0)
sage: vf = E.vector_field([x*y, x^2, z**2])
sage: M = Matrix(RR, 3, 3); M[:] = 1
sage: A = E.tensor_field(1, 1, M)
sage: A
Tensor field of type (1,1) on the Euclidean space E^3
sage: A[:]
[1 1 1]
[1 1 1]
[1 1 1]
sage: A(vf)
Vector field on the Euclidean space E^3
sage: A(vf).display()
(x^2 + x*y + z^2) e_x + (x^2 + x*y + z^2) e_y + (x^2 + x*y + z^2) e_z
sage: A(vf)[:]
[x^2 + x*y + z^2, x^2 + x*y + z^2, x^2 + x*y + z^2]


Note that the action of the endomorphism A onto the vector field vf is obtained by A(vf) and not by A*vf, which would perform a tensor product and would yield a tensor field of type (2, 1).

Note also that if the matrix M is invertible, you may perform the conversion simply as

A = E.automorphism_field(M)

more

Thank you very much. This really helped and saved me from a lot of trouble.

( 2021-01-29 14:26:30 +0200 )edit