Yeah, bra/ket but not quite. I was trying to match your notation where ⟨⋅,⋅⟩ is the expander and [⋅,⋅] is the contractor. I think I mixed up the bracket directions, and it makes more sense as
h=⟨[h11⟩⋯[h1n⟩⋮⋱⋮[hn1⟩⋯[hnn⟩⎤⎥
⎥⎦
that way we can think of it as one big contraction
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 hj,i entries. And in the Rm→Rn case, they’re reals, not row or column vectors.
It is true that you can decompose as either ⟨[…]…[…]⟩ or [⟨…⟩…⟨…⟩]. 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.
Yeah, bra/ket but not quite. I was trying to match your notation where ⟨⋅,⋅⟩ is the expander and [⋅,⋅] is the contractor. I think I mixed up the bracket directions, and it makes more sense as
h=⟨[h11⟩⋯[h1n⟩⋮⋱⋮[hn1⟩⋯[hnn⟩⎤⎥ ⎥⎦
that way we can think of it as one big contraction
[h11⟩,h12⟩,…,hnn⟩]
or expansion
⟨[h11,[h12,…,[hnn⟩
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 hj,i entries. And in the Rm→Rn case, they’re reals, not row or column vectors.
It is true that you can decompose as either ⟨[…]…[…]⟩ or [⟨…⟩…⟨…⟩]. 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.