Dot product for lists

Suppose we want to get the dot product for the following lists aa and bb, whose elements (a, b, etc.) are also lists. We can get the symbolic expression of aa.bb first, and then assign numerical values of the elements in aa and bb.

aa = {{a, b}, {c, d}};
bb = {{{a, b}, {c, d}}, {{a, b}, {c, d}}};
cc = aa.bb;
(* -> {{{a^2 + a b, a b + b^2}, {a c + b c, a d + b d}}, {{a c + a d, b c + b d}, {c^2 + c d, c d + d^2}}} *)
a = b = c = d = {1, 2, 3}
cc
(* -> {{{{2, 8, 18}, {2, 8, 18}}, {{2, 8, 18}, {2, 8, 18}}}, {{{2, 8, 18}, {2, 8, 18}}, {{2, 8, 18}, {2, 8, 18}}}} *)


However, if we do not want to get the symbolic expression for aa.bb (because it is too long in some situations), and only use numerical calculations, we cannot directly use Dot in the same way as follows. Is there a simple way to fix this?

aa = {{a, b}, {c, d}};
bb = {{{a, b}, {c, d}}, {{a, b}, {c, d}}};
a = b = c = d = {1, 2, 3};
cc = aa.bb
(* -> Dot::dotsh: Tensors {{{1,2,3},{1,2,3}},{{1,2,3},{1,2,3}}} and {{{{1,2,3},{1,2,3}},{{1,2,3},{1,2,3}}},{{{1,2,3},{1,2,3}},{{1,2,3},{1,2,3}}}} have incompatible shapes. >> *)

• Have you seen Inner[]? – J. M.'s torpor Jun 19 '15 at 15:21
• Dot works as you desire only when the dimensions are compatible; for example, with a=b=c=d={1,2}. – Ian Jun 19 '15 at 15:26
• Related: Easy way to perform multiplication of two 2x2 matrices, that contain list elements?, but in a sense answer there uses kind of intermediate symbolic representation, so it might not be a best answer to this question. – jkuczm Jun 20 '15 at 9:46
• @Guesswhoitis. Can this be done with inner? I thought so too, but cant make it work. – george2079 Jun 20 '15 at 13:35

not pretty but there is always this:

 Table[Sum[aa[[i, j]] bb[[j, k, l ]],{j,2}],{i,2},{k,2},{l,2}]


If the entire symbolic product is too large perhaps it is sufficient to work in chunks:

aa = {{a, b}, {c, d}};
bb = {aa, aa};

rule = a | b | c | d -> {1, 2, 3};
Table[x.y /. rule, {x, aa}, {y, bb\[Transpose]}]

{{{{2, 8, 18}, {2, 8, 18}}, {{2, 8, 18}, {2, 8, 18}}},
{{{2, 8, 18}, {2, 8, 18}}, {{2, 8, 18}, {2, 8, 18}}}}