programjames comments on Matrices map between biproducts

[programjames]

I believe

$$h=⟩\begin{array}{ccc}{f}_{11}⟩[g_{11}|& \cdots & f_{1n}⟩[g_{1n}|\\ ⋮& \ddots & ⋮\\ f_{n1}⟩[g_{n1}|& \cdots & f_{nn}⟩[g_{nn}|\end{array}[$$

is the main idea. $[g_{ij}|$ (row/​down vector) contracts many values down to one while $|f_{ji}⟩$ (column/​up vector) expands one value back up to many, linearly.

[jessicata]

I don’t understand the notation; it looks like bra/​ket except not quite?

[programjames]

Yeah, bra/​ket but not quite. I was trying to match your notation where $⟨\cdot ,\cdot ⟩$ is the expander and $[\cdot ,\cdot ]$ is the contractor. I think I mixed up the bracket directions, and it makes more sense as

$$h=⟨\begin{array}{ccc}[h_{11}⟩& \cdots & [h_{1n}⟩\\ ⋮& \ddots & ⋮\\ [h_{n1}⟩& \cdots & [h_{nn}⟩\end{array}]$$

that way we can think of it as one big contraction

$$[h_{11}⟩,h_{12}⟩,\dots ,h_{nn}⟩]$$

or expansion

$$⟨[h_{11},[h_{12},\dots ,[h_{nn}⟩$$

[jessicata]

I believe bra/​ket is for row and column vectors. I don’t think it applies here, because in the general case (semiadditive categories), you have arbitrary linear maps as the $h_{j,i}$ entries. And in the $R^m\to R^n$ case, they’re reals, not row or column vectors.

It is true that you can decompose as either $⟨[\dots ]\dots [\dots ]⟩$ or $[⟨\dots ⟩\dots ⟨\dots ⟩]$. To be clear I’m using $⟨⟩$ and [] from category theory product/​coproduct notation, it’s not meant to match linear algebra or bra/​ket notation.