# Overwrite a tensor product notation a_\[TensorProduct]b_ := KroneckerProduct[a, b]

I noticed that the definition of KroneckerProduct and

For example

TensorProduct[({
{0, 1},
{1, 0}
}), ({
{0, 1},
{1, 0}
})]


Output: {{{{0,0},{0,0}},{{0,1},{1,0}}},{{{0,1},{1,0}},{{0,0},{0,0}}}}

KroneckerProduct[({
{0, 1},
{1, 0}
}), ({
{0, 1},
{1, 0}
})]


Output: {{0, 0, 0, 1}, {0, 0, 1, 0}, {0, 1, 0, 0}, {1, 0, 0, 0}}

Mathematically, what we really want for a tensor product between two matrices is indeed the KroneckerProduct, NOT TensorProduct.

I would like to define such a function:

a_\[TensorProduct]b_ := KroneckerProduct[a, b]


But it outputs a failure:

Outputs: SetDelayed::write: Tag TensorProduct in a_[TensorProduct]b_ is Protected.

Is there a way to use the $$\otimes$$ namely [TensorProduct] notation to define the function a_[TensorProduct]b_ := KroneckerProduct[a, b]?

See:

This works for me

CircleTimes[x_, y_] := KroneckerProduct[x, y]


or, as @CarlWoll suggested, simply

CircleTimes = KroneckerProduct


Now

a = {{0, 1}, {1, 0}};
a\[CircleTimes]a
(* {{0, 0, 0, 1}, {0, 0, 1, 0}, {0, 1, 0, 0}, {1, 0, 0, 0}} *)


Notice that it is entered as is entered as \esc c* \esc and not as \esc t* \esc. The latter is indeed reserved.

• thanks but the reason to use the ⊗ symbol is to simplify the notation when we have $$a ⊗ b ⊗ c ⊗d ⊗ e ⊗ f ⊗ ...$$ a long list. Othewise, how would we not just use the $$\text{KroneckerProduct[a, KroneckerProduct[b,KroneckerProduct[c,...] }$$ that is also equally long as your new function.... Aug 20, 2021 at 18:44
• @wonderich See the documentation for CircleTimes, the very first example demonstrates exactly what you want. You may also need to use parenthesis. Aug 20, 2021 at 18:45
• thanks for the comment +1, but I really hope to use a symbol ⊗ or something else very short not like CircleTimes. But it is good to know CircleTimes[x,y, ...] displays as x[CircleTimes]y[CircleTimes].... Aug 20, 2021 at 18:47
• @wonderich As I understood, you would like to handle multiple arguments. Then you have to define corresponding rules or use parenthesis as (a\[CircleTimes]a)\[CircleTimes]a Aug 20, 2021 at 18:49
• Could also do CircleTimes = KroneckerProduct Aug 20, 2021 at 18:58

For display purposes we can define a new display format for KroneckerProduct like this

Unprotect[KroneckerProduct];
Format[x_KroneckerProduct] := Row[List @@ x,
Subscript["\[CircleTimes]", Style["k", Tiny]]]
Protect[KroneckerProduct];

KroneckerProduct[a, b, c, d]


$$a\otimes _{\text{k}}b\otimes _{\text{k}}c\otimes _{\text{k}}d$$

The little "k" subscript on the $$\otimes$$ is there to make it a unique symbol, so there can be no question about its meaning. It is not a required feature.

To save a few keystrokes and to make our input fit the language of tensor products we could define an abbreviation, tensor, as

ClearAll[tensor]
tensor = KroneckerProduct;
expr = tensor[a, b, c, d]


We might read this as "a tensored with b tensored with c tensored with d", or "tensor product of a, b, c and d". expr displays with $$\otimes_k$$ symbols, but is only a KroneckerProduct expression, as can be seen with FullForm.

To remove the unique symbol definition and return KroneckerProduct to its usual display form, evaluate this

Unprotect[KroneckerProduct];
Format[_KroneckerProduct] = Null;
Format[_KroneckerProduct] =.
Protect[KroneckerProduct];


The assignment of Null in the second line ensures there is a value to unset in the third line.