Dimensions[] not working with assumptions?

Question

I am trying to perform some symbolic calculus with tensors and matrices. I have recently noticed that one can set assumptions on variables by using the command $Assumptions$ as described here. However, when I make

$Assumptions$ = U ∈ Matrices[{d, m}, Reals];
$Assumptions$ = P ∈ Matrices[{d, t}, Reals];


Y = Transpose[U].P;
Dimensions[Y]

I get

(* {2} *)

Using TensorDimensions doesn't return a useful result either, as

TensorDimensions[Y]

(* TensorDimensions[Transpose[U].Phi] *)

If I am assuming $U$ to be a $d\times m$ matrix and $P$ to be a $d\times t$ matrix, then the product between $U^{\intercal}P$ should be an $m\times t$ matrix. I am sure I am probably misunderstanding either the usage of $Assumptions$ or the usage of the command Dimensions[] regarding this particular case. But if that is so, why is it not working this way?

Answer 1

You just have a small typo in $Assumptions and this works only with TensorDimensions

$Assumptions = {U ∈ Matrices[{d, m}, Reals], 
   P ∈ Matrices[{d, t}, Reals]};
Y = Transpose[U].P;
Y // Dimensions
Y // TensorDimensions

{2}

{m, t}