Dear all,

I am looking for the symbolic result of matrix product A.P.Transpose[A] while working with block matrix.

In the current version of the notebook (see below), Mathematica assumes the elements of the matrices as scalar and therefore output the result under scalar form and using commutative properties of the scalar (while it is not applicable when working with matrices)

A = ( {
    {ABgBg, 0, 0, 0, 0, 0},
    {0, ASgSg, 0, 0, 0, 0},
    {0, 0, AMgMg, 0, 0, 0},
    {0, 0, 0, ABaBa, 0, 0},
    {ATeBg, ATeSg, ATeMg, 0, ATeTe, ATeDv},
    {0, 0, 0, ADvBa, ADvTe, ADvDv}
   } );
P = ( {
    {PBgBg, PBgSg, PBgMg, PBgBa, PBgTe, PBgDv},
    {PBgSg\[Transpose], PSgSg, PSgMg, PSgBa, PSgTe, PSgDv},
    {PBgMg\[Transpose], PSgMg\[Transpose], PMgMg, PMgBa, PMgTe, PMgDv},
    {PBgBa\[Transpose], PSgBa\[Transpose], PMgBa\[Transpose], PBaBa, 
     PBaTe, PBaDv},
    {PBgTe\[Transpose], PSgTe\[Transpose], PMgTe\[Transpose], 
     PBaTe\[Transpose], PTeTe, PTeDv},
    {PBgDv\[Transpose], PSgDv\[Transpose], PMgDv\[Transpose], 
     PBaDv\[Transpose], PTeDv\[Transpose], PDvDv}
   } );

Res = A.P.A\[Transpose];
Res // MatrixForm

However, I would like to get the symbolic result of A.P.Transpose[A] but doing assumptions that each elements of the matrix A and P are themselves a matrix of dimensions {3,3}

For instance, with the current notebook the output of APAT[[1,1]] is ABgBg² PBgBg while it should be ABgBg.PBgBgTranspose[ABgBg].

I tried to add assumptions (see below) but the product A.P.Transpose[A] results in Tensor errors...

$Assumptions = 
  Element[ABgBg | ABgBg | ASgBg | ASgSg | PBgBg | PBgSg | PSgSg, 
   Matrices[{3, 3}, Reals]];

Thanks for your help and do not hesitate if not clear.