Does matrix multiplication require an inner product space?

Question

Does matrix multiplication require an inner product space?

It would seem to me that since multiplying a vector by a matrix is simply a linear map, and multiplying two matrices the composition of a linear maps, these must be defined in any vector space, and should not depend on an inner product.

Yet, any definition of matrix multiplication I can consider immediately induces an inner product. I understand that you can multiply a vector by a matrix without using an inner product, but, once you do so, you immediately define an inner product.

So: Can matrix multiplication exist in vector spaces without inner products? What is the relationship between the two?

Answer 1

The thing is, to translate linear maps into matrices, you need to choose a basis.

But when you choose a basis (say $(e_1,\dots,e_n)$), you also automatically get a canonical bilinear form given by $$x\cdot y = \sum x_iy_i$$ where the $x_i$ and $y_i$ are the coordinates of $x$ and $y$ in the fiven basis.

So, indeed matrix multiplication has nothing to do with inner products, but in any context where you can represent linear endomorphisms by matrices, you automatically also have an inner product (rather, a bilinear form, which is an inner product if you work with real numbers).

Answer 2

Let $A\in M_{m×n}(\Bbb{K}),B\in M_{n×p}(\Bbb{K})$.

Then consider the linear map associated with $A$ , $$L_A:\Bbb{K}^n\to\Bbb{K}^m$$ defined by $L_A(X) =AX$

Similarly consider the linear map associated with $B$ , $$L_B:\Bbb{K}^p\to\Bbb{K}^n$$ defined by $L_B(X) =BX$

Then $AB=L_A\circ L_B$

Composition of two linear maps $T:U\to V$ and $S:V\to W$,$S\circ T$ is always defined where $U, V, W$ any three vector spaces over the field $\Bbb{K}$.