What is the name of the algebra that is under the multiplication of matrices of different sizes?

I understand that Matrix Group(Linear Group) is a group for invertible square matrices under multiplication, Matrix Ring is a ring for square matrices under both multiplication and addition. But they are all limited to special kinds of matrices. What I'm looking for is the algebra for multiplication of matrices of different sizes. Can I have the name of it please?

• How do you multiply a $2\times 2$-matrix by a $3\times 3$-matrix "all the time in Linear Algebra"? – Dietrich Burde May 13 '20 at 14:34
• The set of all matrices of any size does not have the nice property that you can add any pair of matrices together or multiply any pair of matrices together, something that is highly desirable, and so the set of all matrices does not count as a ring or even a monoid or group. It is not even a set with a well-defined binary operation. You can only add matrices of the same shape and you can only multiply matrices where the number of rows of the one is the number of columns of the other. – JMoravitz May 13 '20 at 14:36
• I think you are looking for an algebraic structure that can describe matrix multiplication of matrices of different sizes, provided that they match. Then you are looking for a category. – Jens Renders May 13 '20 at 14:40
• @DietrichBurde I'm sorry for the poor articulation(English is my 2nd lang). I'm aware of the condition for multiplication in linear algebra. What I was thinking was since it's a product of matrices, and a product is usually from an algebra, I thought the product must has been from some kind of algebra. – Ingun전인건 May 13 '20 at 14:41
• @Ingun전인건 You are right, we call it a product and it is indeed the operation of an algebraic structure. I think you actually came up with a very good example that shows that a category is an algebraic structure just like a group or a ring. It is just a little bit more complex ;) – Jens Renders May 13 '20 at 15:03

The algebraic structure you are looking for is called a category.

The easiest way to see this is that matrices (with coefficients in for example $$\mathbb{R}$$) are in bijective correspondence with linear maps between finite dimensional vector spaces over $$\mathbb{R}$$. So this algebraic structure you are thinking about is simply a category of finite dimensional vector spaces over $$\mathbb{R}$$. Matrices are the morphisms of this category, and the matrix multiplication is the composition operation.

Let's be more precise. To define a category (let's call it $$\mathcal{C}$$), we have to provide a set of objects, for wich we will take $$\operatorname{obj}(\mathcal{C})=\{\mathbb{R}^n \mid n \in \mathbb{N} \}$$ We also have to provide a set of morphisms for each pair of objects. So for the pair $$(\mathbb{R}^n, \mathbb{R}^m)$$ we take the set of all linear maps between thes 2 spaces, which is equivalent to the set of all $$m$$ by $$n$$ matrices. $$\operatorname{Hom}_{\mathcal{C}}(\mathbb{R}^n, \mathbb{R}^m) = \mathbb{R}^{m\times n}$$

In a category you then need to define a composition operator, which is in our case the matrix multiplication or equivalently the composition of linear maps.

You can check for yourself that this indeed satisfies the necessary properties to be a category (associativity and identity).

We can go further: there is also an algebraic structure that describes both the addition and multiplication of general matrices. This structure is called an abelian category.

• @Ingun전인건 The point is that a category is an algebraic structure, just like a group or a ring. I know this can be confusing, because all groups/rings together form a category, but categories themselves are simple algebraic things (that also form a category together). Take your time to parse my answer, or even try to invent the algebraic structure yourself, then you will reinvent the structere that we call a category. – Jens Renders May 13 '20 at 15:10
• @Ingun전인건 Take a look at the general definition of a category, and then try to see how the composition operator is just like a multiplication in a group or monoid. It just has the added restrictions that it cannot be applies to any 2 elements, just like matrix multiplication. – Jens Renders May 13 '20 at 15:12
• @Ingun전인건 This might also help: What if we take the definition of a category, but we only allow one object. This is just what a monoid is! can you see this? And if we then demand that each morphism is invertible, then we have defined a group! – Jens Renders May 13 '20 at 15:15
• @Ingun전인건 The definition of algebraic structure that they are using in that article is quite restrictive. For example, a groupoid also does not fit that description. But I think you are still confused by the fact that we have a category of all groups. Did you know that we also have a category of categories? Does that clear something up? I am not talking about categories that have algebraic structures as objects, I am talking about categories as objects themselves! Difficult to wrap your head around I know. – Jens Renders May 13 '20 at 15:52
• @Ingun전인건 If you can understand how a category with one object is exactly the same thing as a monoid, then you will be getting there. Focus on that first, then try to think of a group as a category with one object with extra properties. – Jens Renders May 13 '20 at 15:54