Matrix Multiplication

Definition Let $A$ be an $m \times n$ matrix and $B$ a $p \times q$ matrix. The product $AB$ is defined exactly when $n = p$, that is, when $A$ has as many columns as $B$ has rows. In that case $AB$ is an $m \times q$ matrix,

Definition Let $A = [a_{ij}]$ be $m \times n$ and $B = [b_{ij}]$ be $n \times q$. The product $AB$ is the $m \times q$ matrix whose entry in row $i$ and column $j$ is

This is the dot product of row $i$ of $A$ with column $j$ of $B$, both read as vectors with $n$ components.

Definition Write a vector with $n$ real components as an $n \times 1$ column, so $\langle a, b \rangle$ becomes the column with entries $a$ and $b$. The set of all such vectors is $\mathbb{R}^n$, and a point stands for its position vector. An $m \times n$ matrix $A$ defines the function $\mathbf{x} \mapsto A\mathbf{x}$ from $\mathbb{R}^n$ to $\mathbb{R}^m$, where $A\mathbf{x}$ is the matrix product. This function is linear, meaning

for all $\mathbf{u}$ and $\mathbf{v}$ in $\mathbb{R}^n$ and every real number $c$. Conversely, every linear map from $\mathbb{R}^n$ to $\mathbb{R}^m$ has this form for exactly one $m \times n$ matrix, the one whose column $k$ is the image of $\mathbf{e}_k$, the vector with $1$ in place $k$ and $0$ everywhere else. For maps the word linear has this stricter meaning. With one variable, $x \mapsto mx$ is a linear map, but a linear function $mx + b$ with $b \ne 0$ is not, since it sends $0$ to $b$.

Definition Let $A$ be $m \times n$ and $B$ be $n \times p$. For every $\mathbf{x}$ in $\mathbb{R}^p$

So applying $B$ first and then $A$ is the same linear map as applying the single matrix $AB$. The matrix that acts first stands on the right. The same reasoning shows that multiplication is associative, $(AB)C = A(BC)$ whenever the sizes fit.

Definition Matrix multiplication is not commutative. The product $AB$ can be defined while $BA$ is not. When both are defined they can have different sizes, and for every $n \ge 2$ there are $n \times n$ matrices with $AB \ne BA$. Some pairs do commute, but that has to be checked, never assumed. Rules for numbers that rely on $ab = ba$ can fail. For square matrices $A$ and $B$ of the same size

which equals $A^2 + 2AB + B^2$ only when $AB = BA$.

Definition The $n \times n$ identity matrix $I_n$ has every entry on the main diagonal, the entries in row $i$ and column $i$, equal to $1$ and every other entry equal to $0$. For every $m \times n$ matrix $A$

As a map, $I_n \mathbf{x} = \mathbf{x}$ for every $\mathbf{x}$ in $\mathbb{R}^n$. In particular $I_n$ commutes with every $n \times n$ matrix.

Definition The $m \times n$ zero matrix $O$ has every entry equal to $0$. For every $m \times n$ matrix $A$ we have $A + O = A$, and whenever the sizes fit, a product with a zero factor is a zero matrix, $AO = O$ and $OB = O$. Unlike numbers, two nonzero matrices can have product $O$. So $AB = O$ does not force $A = O$ or $B = O$, and $AB = AC$ does not force $B = C$.