The Category of Matrices II
We define the monoidal product on objects by multiplication — — and on morphisms by using the Kronecker product. That is, if we have an matrix and an matrix , then we get the Kronecker product
Here we have to be careful about what we’re saying. In accordance with our convention, the pair of indices (with and ) should be considered as the single index . It’s clear that this quantity then runs between and . A similar interpretation goes for the index pairs .
Of course, we need some relations for this to be a monoidal structure. Strict associativity is straightforward:
For our identity object, we naturally use , with its identity morphism . Note that the first of these is the object the natural number , while the second is the matrix whose single entry is the field element . Then we can calculate the Kronecker product to find
and so strict associativity holds as well.
The category of matrices also has duals. In fact, each object is self-dual! That is, we set . We then need our arrows and .
The morphism will be a matrix. Specifically, we’ll use , with and both running between and . Again, we interpret an index pair as described above. The symbol is another form of the Kronecker delta, which takes the value when its indices agree and when they don’t.
Similarly, will be an matrix: , using yet another form of the Kronecker delta.
Now we have compatibility relations. Since the monoidal structure is strict, these are simpler than usual:
But now all the basic matrices in sight are various Kronecker deltas! The first equation reads
which is true. You should be able to verify the second one similarly.
The upshot is that we’ve got the structure of a monoidal category with duals on .