3D matrix "contraction"

[Michele Esposito]

Hello everyone,

I'm new in this group, I got to it via this link "3 Dimensional Matrix Multiplication - MAGMA Forum (utk.edu)".

I try to explain what I am trying to achieve.

NOTE: I am NOT a mathematician. I say "contraction" for reasons that will probably be clear in a few lines, but I am not sure it is the right proper mathematical definition of such an operation.

I have two basic inputs which I need to combine, depending on the context:

 - "t", vector of dimensions "D x 1";

 -  "A" matrix of some coefficients of dimension "D x L".

They are the basic for defining a vector "f" of dimensions "L x 1" via

     f = A^T  .  t

Now, for simplicity, I start from the 2D case. For some reasons, I want to compute  <f  .  f^T> (matrix of dimensions "L x L"), which is given by

     f  f^T  =  A^T  .  t   .  t^T  .   A

in which

     T   =   t   .   t^T

is a matrix of dimensions "D x D".

NOTE: now should be clear why I think to  "contraction". Latter equations give a "L x L" matrix from a "D x D" one, in which usually   L < D.

Now, I would like to extend this concept to the 3D case. What I have in mind is that I will have a 3D matrix of dimensions "D x D x D"  (built from the only input  "t") which is finally compacted into a "L x L x L" matrix using the other known input, the matrix A.

I studied a bit of tensor calculus, but I am not sure how this is of application in this case.

Kind regards,

Michele