## Matrices III

Given two finite-dimensional vector spaces and , with bases and respectively, we know how to build a tensor product: use the basis .

But an important thing about the tensor product is that it’s a *functor*. That is, if we have linear transformations and , then we get a linear transformation . So what does *this* operation look like in terms of matrices?

First we have to remember exactly how we get the tensor product . Clearly we can consider the function . Then we can compose with the bilinear function to get a bilinear function from to . By the universal property, this must factor uniquely through a linear function . It is this map we call .

We have to pick bases of and of . This gives us a matrix coefficients for and for . To calculate the matrix for we have to evaluate it on the basis elements of . By definition we find:

that is, the matrix coefficient between the index pair and the index pair is .

It’s not often taught anymore, but there is a name for this operation: the Kronecker product. If we write the matrices (as opposed to just their coefficients) and , then we write the Kronecker product .

[…] Like we saw with the tensor product of vector spaces, the dual space construction turns out to be a functor. In […]

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[…] is one slightly touchy thing we need to be careful about: Kronecker products. When the upper index is a pair with and we have to pick an order on the set of such pairs. […]

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[…] the monoidal product on objects by multiplication — — and on morphisms by using the Kronecker product. That is, if we have an matrix and an matrix , then we get the Kronecker […]

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[…] wait, there’s more! The functor is linear over , so it’s a functor enriched over . The Kronecker product of matrices corresponds to the monoidal product of linear transformations, so the functor is […]

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[…] can recognize this as a Kronecker product of two […]

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[…] want. And we know that when expressed in matrix form, the tensor product of linear maps becomes the Kronecker product of matrices. We write the character of as , that of as , and that of their tensor product as , […]

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