The Unapologetic Mathematician

Mathematics for the interested outsider

The Category of Matrices II

As we consider the category \mathbf{Mat}(\mathbb{F}) of matrices over the field \mathbb{F}, we find a monoidal structure.

We define the monoidal product \boxtimes on objects by multiplication — m\boxtimes n=mn — and on morphisms by using the Kronecker product. That is, if we have an m_1\times n_1 matrix \left(s_{i_1}^{j_1}\right)\in\hom(n_1,m_1) and an m_2\times n_2 matrix \left(t_{i_2}^{j_2}\right)\in\hom(n_2,m_2), 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 (i_1,i_2) (with 1\leq i_1\leq m_1 and 1\leq i_2\leq m_2) should be considered as the single index (i_1-1)m_2+i_2. It’s clear that this quantity then runs between {1} and m_1m_2. A similar interpretation goes for the index pairs (j_1,j_2).

Of course, we need some relations for this to be a monoidal structure. Strict associativity is straightforward:


For our identity object, we naturally use {1}, with its identity morphism \left(1\right). Note that the first of these is the object the natural number {1}, while the second is the 1\times1 matrix whose single entry is the field element {1}. 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 n^*=n. We then need our arrows \eta_n:1\rightarrow n\boxtimes n and \epsilon_n:n\boxtimes n\rightarrow1.

The morphism \eta_n will be a 1\times n^2 matrix. Specifically, we’ll use \eta_n=\left(\delta^{i,j}\right), with i and j both running between {1} and n. Again, we interpret an index pair as described above. The symbol \delta^{i,j} is another form of the Kronecker delta, which takes the value {1} when its indices agree and {0} when they don’t.

Similarly, \epsilon_n will be an n^2\times1 matrix: \epsilon_n=\left(\delta_{i,j}\right), 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 \mathbf{Mat}(\mathbb{F}).

June 3, 2008 Posted by | Algebra, Category theory, Linear Algebra | 1 Comment