what is the geometry behind the matrix multiplication?

Question

What is the geometry behind the matrix multiplication?

The questions that I am having is the follows.

$\bullet$ I accept that we are viewing $\mathbb R^4$ as either in $\begin{pmatrix} a_{11},a_{12},a_{13},a_{14}\end{pmatrix}$ or $\begin{pmatrix} a_{11}&a_{12}\\a_{13}&a_{14}\end{pmatrix}$. so, matrix addition makes sense that it gives another vector in that space.

$\bullet$ But, Matrix multiplication does not convince me in this role.

I strucked with,

Like in the matrix addition, (addition of two vectors is nothing but the diagonal of the parallelogram in which the two vectors are adjustcent sides)

is there any vector space diagramatic representation for matrix multiplication??

Note: I am aiming to teach this factacy to my grade 11 students who are studying their matrices now only.(Means to say, this is the first time they gonna meet matrices)

Answer 1

Another perspective is to use the SVD, hence a matrix multiplication can be thought of as a rotation, followed by a (per dimension) scaling, followed by another rotation.

That is any (real) matrix $A$ can be written as $A=U \Sigma V^T$, where $U,V$ are rotations and $\Sigma$ is diagonal, with non negative entries (in decreasing order on the diagonal).

Answer 2

The "geometry" behind matrix multiplication is the composition of the corresponding linear maps. The matrices $A\in M(n\times m, \mathbb k)$ and $B \in M(m \times p, \mathbb k)$ define with respect to the canonical bases linear maps $f_B: \mathbb k^p \longrightarrow \ \mathbb k^m$ and $f_A: \mathbb k^m \longrightarrow \ \mathbb k^n$. Similar, the product $C:=AB \in M(n\times p, \mathbb k)$ defines a linear map $f_C: \mathbb k^p \longrightarrow \ \mathbb k^n$. One proves that the linear map $f_C$ is the composition $f_C = f_A \circ f_B$. Here $\mathbb k$ denotes the base field, e.g., $\mathbb k = \mathbb R$.

Answer 3

There are several points of view to interpret matrix multiplication. Let's say you're doing $A B = C$ where $A$ is $m \times n$ and $B$ is $n \times p$, so $C$ is $m \times p$.

1. Each column of $C$ is the matrix-vector product of $A$ with the corresponding column of $B$. Thus the $j$'th column of $C$ is a linear combination of the columns of $A$ with coefficients given by the entries of the $j$'th column of $B$.
2. Each row of $C$ is the vector-matrix product of the corresponding row of $A$ with the matrix $B$. Thus the $i$'th row of $C$ is a linear combination of the rows of $B$, with coefficients given by the entries of the $i$'th row of $A$.