Contraction of square tensors

Question

Let there be tensors A and B

A = Outer[Times, {1, 0}, {2, 0}]
B = Grad[{f[x, y], g[x, y]}, {x, y}]

with output

{{2, 0}, {0, 0}}
{{(f^(1,0))[x,y],(f^(0,1))[x,y]},{(g^(1,0))[x,y],(g^(0,1))[x,y]}}

Now, I am looking for tensor contraction of A and B ($A:B$) as follows

TensorContract[A, B]

Which produces output

TensorContract::contr: Invalid contraction {(f^(1,0))[x,y],(f^(0,1))[x,y]}.

How can get the correct result, in this case $2\frac{\partial f}{\partial x}$?

Answer 1

Simplest method for matrices:

Total[A B, 2]

One can also use a Frobenius innerproduct

Tr[Transpose[A].B]

or simply

Flatten[A].Flatten[B]

In fact, the last one should be the most efficient for large numerical matrices (it takes advantage of vectorization and fused multiply-add operations).

If one insists on using TensorContract, one has to generate a tensor $A \otimes B$ first and then contract the slot pairs {1, 3} and {2, 4}

TensorContract[TensorProduct[A, B], {{1, 3}, {2, 4}}]

This is however not a good idea, because the intermediate tensor $A \otimes B$ contains $n^4$ elements when $A$ and $B$ are both $n \times n$ matrices.

By the way, Tr[Transpose[A].B] is also of complexity $O(n^3)$ due to the matrix-matrix product; so better also avoid that one.