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

\left(s_{i_1}^{j_1}\right)\boxtimes\left(t_{i_2}^{j_2}\right)=\left(s_{i_1}^{j_1}t_{i_2}^{j_2}\right)

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:

\left(\left(r_{i_1}^{j_1}\right)\boxtimes\left(s_{i_2}^{j_2}\right)\right)\boxtimes\left(t_{i_3}^{j_3}\right)=\left((r_{i_1}^{j_1}s_{i_2}^{j_2})t_{i_3}^{j_3}\right)=\left(r_{i_1}^{j_1}(s_{i_2}^{j_2}t_{i_3}^{j_3})\right)=\left(r_{i_1}^{j_1}\right)\boxtimes\left(\left(s_{i_2}^{j_2}\right)\boxtimes\left(t_{i_3}^{j_3}\right)\right)

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

\left(t_i^j\right)\boxtimes\left(1\right)=\left(t_i^j\right)=\left(1\right)\boxtimes\left(t_i^j\right)

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:

(\epsilon_n\otimes1_n)\circ(1_n\otimes\eta_n)=1_n
(1_{n^*}\otimes\epsilon_n)\circ(\eta_n\otimes1_{n^*})=1_{n^*}

But now all the basic matrices in sight are various Kronecker deltas! The first equation reads

\left(\delta_a^b\delta^{c,d}\right)\left(\delta_{b,c}\delta_d^e\right)=\delta_a^e

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

   

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