2 fixed a typo

Here's how I live to think about matrices. Penrose (1971) figured out that you can draw linear algebra diagrammatically. A picture in the Penrose notation is a directed labeled graph with external leaves. The edges are labeled by vector spaces (changing the direction on an edge has the same effect as swapping the label X with the dual vector space X*), and vertices by multilinear maps. In this way, placing two edges next to each other is the tensor product. The ground field R should be drawn as an invisible edge, so that XR = X.

So, pick your favorite finite-dimensional vector space X, and think about the types of diagrams you can draw using just it. Well, the space of matrices (what you call V) is XX*, so it looks like two parallel lines pointed in opposite directions. Then you can check that the trace is the directed cap, the identity element (thought of as a map RV) is the directed cup, and multiplication and comultiplication are both given by trivalent vertices.

In ASKII ASCII (ignore the weird coloring):

       |            |                        | |
|            |                        | |
X  =  ^ ,   X*  =  v ,  R = [empty],  V  =  ^ v
|            |                        | |
|            |                        | |

->-            |     |
/   \           ^     v
Tr  =  |     |    I  =  |     |
^     v           \   /
|     |            -<-

| |                  | |   | |
^ v                  ^ v   ^ v
| |                  | |   | |
mu  =     / _ \       delta  =   \ \_/ /
/ / \ \                  \   /
| |   | |                  | |
^ v   ^ v                  ^ v
| |   | |                  | |


Not only does the notation "explain" the comultiplication, it "proves" all the associativity and unital properties you might want. Mostly, though, I think it makes it totally clear what the Frobenius pairing (a,b) → Tr(ab) is doing. It's just the map:

              ->-
/ _ \
pair =     / / \ \
| |   | |
^ v   ^ v
| |   | |


Which is just the canonical fact that (XX*)* = XX*. This ability to rotate XX* is why δ = μ*.

1

Here's how I live to think about matrices. Penrose (1971) figured out that you can draw linear algebra diagrammatically. A picture in the Penrose notation is a directed labeled graph with external leaves. The edges are labeled by vector spaces (changing the direction on an edge has the same effect as swapping the label X with the dual vector space X*), and vertices by multilinear maps. In this way, placing two edges next to each other is the tensor product. The ground field R should be drawn as an invisible edge, so that XR = X.

So, pick your favorite finite-dimensional vector space X, and think about the types of diagrams you can draw using just it. Well, the space of matrices (what you call V) is XX*, so it looks like two parallel lines pointed in opposite directions. Then you can check that the trace is the directed cap, the identity element (thought of as a map RV) is the directed cup, and multiplication and comultiplication are both given by trivalent vertices.

In ASKII (ignore the weird coloring):

       |            |                        | |
|            |                        | |
X  =  ^ ,   X*  =  v ,  R = [empty],  V  =  ^ v
|            |                        | |
|            |                        | |

->-            |     |
/   \           ^     v
Tr  =  |     |    I  =  |     |
^     v           \   /
|     |            -<-

| |                  | |   | |
^ v                  ^ v   ^ v
| |                  | |   | |
mu  =     / _ \       delta  =   \ \_/ /
/ / \ \                  \   /
| |   | |                  | |
^ v   ^ v                  ^ v
| |   | |                  | |


Not only does the notation "explain" the comultiplication, it "proves" all the associativity and unital properties you might want. Mostly, though, I think it makes it totally clear what the Frobenius pairing (a,b) → Tr(ab) is doing. It's just the map:

              ->-
/ _ \
pair =     / / \ \
| |   | |
^ v   ^ v
| |   | |


Which is just the canonical fact that (XX*)* = XX*. This ability to rotate XX* is why δ = μ*.