## 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 [...]

Pingback by Matrices IV « The Unapologetic Mathematician | May 28, 2008 |

[...] 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. [...]

Pingback by Matrix notation « The Unapologetic Mathematician | May 30, 2008 |

[...] 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 [...]

Pingback by The Category of Matrices III « The Unapologetic Mathematician | June 23, 2008 |

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

Pingback by More Commutant Algebras « The Unapologetic Mathematician | October 5, 2010 |

[...] 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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