Matrix-to-Vector Multipication: Given a $m\times n$ matrix $A$ and a $n$-dimensional vector $x$:

$A=\left[\begin{array}{ccccc}\hfill a_{⟨1,1⟩}\hfill & \hfill a_{⟨1,2⟩}\hfill & \hfill a_{⟨1,3⟩}\hfill & \hfill \cdots \hfill & \hfill a_{⟨1,n⟩}\hfill \\ \hfill a_{⟨2,1⟩}\hfill & \hfill a_{⟨2,2⟩}\hfill & \hfill a_{⟨2,3⟩}\hfill & \hfill \cdots \hfill & \hfill a_{⟨2,n⟩}\hfill \\ \hfill a_{⟨3,1⟩}\hfill & \hfill a_{⟨3,2⟩}\hfill & \hfill a_{⟨3,3⟩}\hfill & \hfill \cdots \hfill & \hfill a_{⟨3,n⟩}\hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill \\ \hfill a_{⟨m,1⟩}\hfill & \hfill a_{⟨m,2⟩}\hfill & \hfill a_{⟨m,3⟩}\hfill & \hfill \cdots \hfill & \hfill a_{⟨m,n⟩}\hfill \end{array}\right]=\left[\begin{array}{c}\hfill {\overrightarrow{a}}_{⟨1,\ast ⟩}\hfill \\ \hfill {\overrightarrow{a}}_{⟨2,\ast ⟩}\hfill \\ \hfill {\overrightarrow{a}}_{⟨3,\ast ⟩}\hfill \\ \hfill ⋮\hfill \\ \hfill {\overrightarrow{a}}_{⟨m,\ast ⟩}\hfill \end{array}\right],\overrightarrow{x}=(x_1,x_2,\cdots ,x_n)$

The result of $A\cdot \overrightarrow{x}$ is an $n$-dimensional vector computed as:

$A\cdot \overrightarrow{x}=({\overrightarrow{a}}_{⟨1,\ast ⟩}\cdot \overrightarrow{x},{\overrightarrow{a}}_{⟨2,\ast ⟩}\cdot \overrightarrow{x},\cdots ,{\overrightarrow{a}}_{⟨m,\ast ⟩}\cdot \overrightarrow{x})=\left(\underset{i=1}{\overset{n}{\sum <!--\; FUNCTION\; APPLICATION\; -->}}{a}_{1,i}\cdot x_i,\sum _{i=1}^n{a}_{2,i}\cdot x_i,\cdots ,\sum _{i=1}^n{a}_{m,i}\cdot x_i\right)$

Matrix-to-Matrix Multiplication: Given a $m\times n$ matrix $A$ and a $n\times k$ matrix $B$:

$A=\left[\begin{array}{ccccc}\hfill a_{⟨1,1⟩}\hfill & \hfill a_{⟨1,2⟩}\hfill & \hfill a_{⟨1,3⟩}\hfill & \hfill \cdots \hfill & \hfill a_{⟨1,n⟩}\hfill \\ \hfill a_{⟨2,1⟩}\hfill & \hfill a_{⟨2,2⟩}\hfill & \hfill a_{⟨2,3⟩}\hfill & \hfill \cdots \hfill & \hfill a_{⟨2,n⟩}\hfill \\ \hfill a_{⟨3,1⟩}\hfill & \hfill a_{⟨3,2⟩}\hfill & \hfill a_{⟨3,3⟩}\hfill & \hfill \cdots \hfill & \hfill a_{⟨3,n⟩}\hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill \\ \hfill a_{⟨m,1⟩}\hfill & \hfill a_{⟨m,2⟩}\hfill & \hfill a_{⟨m,3⟩}\hfill & \hfill \cdots \hfill & \hfill a_{⟨m,n⟩}\hfill \end{array}\right]$, $B=\left[\begin{array}{ccccc}\hfill b_{⟨1,1⟩}\hfill & \hfill b_{⟨1,2⟩}\hfill & \hfill b_{⟨1,3⟩}\hfill & \hfill \cdots \hfill & \hfill b_{⟨1,k⟩}\hfill \\ \hfill b_{⟨2,1⟩}\hfill & \hfill b_{⟨2,2⟩}\hfill & \hfill b_{⟨2,3⟩}\hfill & \hfill \cdots \hfill & \hfill b_{⟨2,k⟩}\hfill \\ \hfill b_{⟨3,1⟩}\hfill & \hfill b_{⟨3,2⟩}\hfill & \hfill b_{⟨3,3⟩}\hfill & \hfill \cdots \hfill & \hfill b_{⟨3,k⟩}\hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill \\ \hfill b_{⟨n,1⟩}\hfill & \hfill b_{⟨n,2⟩}\hfill & \hfill b_{⟨n,3⟩}\hfill & \hfill \cdots \hfill & \hfill b_{⟨n,k⟩}\hfill \end{array}\right]$

The result of $A\cdot B$ is a $m\times k$ matrix computed as:

$A\cdot B=\left[\begin{array}{ccccc}\hfill \underset{i=1}{\overset{n}{\sum <!--\; FUNCTION\; APPLICATION\; -->}}{a}_{⟨1,i⟩}{b}_{⟨i,1⟩}\hfill & \hfill \underset{i=1}{\overset{n}{\sum <!--\; FUNCTION\; APPLICATION\; -->}}{a}_{⟨1,i⟩}{b}_{⟨i,2⟩}\hfill & \hfill \underset{i=1}{\overset{n}{\sum <!--\; FUNCTION\; APPLICATION\; -->}}{a}_{⟨1,i⟩}{b}_{⟨i,3⟩}\hfill & \hfill \cdots \hfill & \hfill \underset{i=1}{\overset{n}{\sum <!--\; FUNCTION\; APPLICATION\; -->}}{a}_{⟨1,i⟩}{b}_{⟨i,k⟩}\hfill \\ \hfill \underset{i=1}{\overset{n}{\sum <!--\; FUNCTION\; APPLICATION\; -->}}{a}_{⟨2,i⟩}{b}_{⟨i,1⟩}\hfill & \hfill \underset{i=1}{\overset{n}{\sum <!--\; FUNCTION\; APPLICATION\; -->}}{a}_{⟨2,i⟩}{b}_{⟨i,2⟩}\hfill & \hfill \underset{i=1}{\overset{n}{\sum <!--\; FUNCTION\; APPLICATION\; -->}}{a}_{⟨2,i⟩}{b}_{⟨i,3⟩}\hfill & \hfill \cdots \hfill & \hfill \underset{i=1}{\overset{n}{\sum <!--\; FUNCTION\; APPLICATION\; -->}}{a}_{⟨2,i⟩}{b}_{⟨i,k⟩}\hfill \\ \hfill \underset{i=1}{\overset{n}{\sum <!--\; FUNCTION\; APPLICATION\; -->}}{a}_{⟨3,i⟩}{b}_{⟨i,1⟩}\hfill & \hfill \underset{i=1}{\overset{n}{\sum <!--\; FUNCTION\; APPLICATION\; -->}}{a}_{⟨3,i⟩}{b}_{⟨i,2⟩}\hfill & \hfill \underset{i=1}{\overset{n}{\sum <!--\; FUNCTION\; APPLICATION\; -->}}{a}_{⟨3,i⟩}{b}_{⟨i,3⟩}\hfill & \hfill \cdots \hfill & \hfill \underset{i=1}{\overset{n}{\sum <!--\; FUNCTION\; APPLICATION\; -->}}{a}_{⟨3,i⟩}{b}_{⟨i,k⟩}\hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill \\ \hfill \underset{i=1}{\overset{n}{\sum <!--\; FUNCTION\; APPLICATION\; -->}}{a}_{⟨m,i⟩}{b}_{⟨i,1⟩}\hfill & \hfill \underset{i=1}{\overset{n}{\sum <!--\; FUNCTION\; APPLICATION\; -->}}{a}_{⟨m,i⟩}{b}_{⟨i,2⟩}\hfill & \hfill \underset{i=1}{\overset{n}{\sum <!--\; FUNCTION\; APPLICATION\; -->}}{a}_{⟨m,i⟩}{b}_{⟨i,3⟩}\hfill & \hfill \cdots \hfill & \hfill \underset{i=1}{\overset{n}{\sum <!--\; FUNCTION\; APPLICATION\; -->}}{a}_{⟨m,i⟩}{b}_{⟨i,k⟩}\hfill \end{array}\right]$

Associative:

$(\mathit{𝐴𝐵})C=A(\mathit{𝐵𝐶})$

$A(\mathit{𝐵𝑥})=(\mathit{𝐴𝐵})x$

Distributive:

$A(x+y)=\mathit{𝐴𝑥}+\mathit{𝐴𝑦}$

$A(x\odot y)=\mathit{𝐴𝑥}\odot \mathit{𝐴𝑦}$

$A⟨x,y⟩=⟨\mathit{𝐴𝑥},\mathit{𝐴𝑦}⟩$

$A⟨⟨x,y⟩⟩=⟨⟨\mathit{𝐴𝑥},\mathit{𝐴𝑦}⟩⟩$

(However, $\mathit{𝐴𝑥}\cdot \mathit{𝐴𝑦}\ne A(x\cdot y)$, because the resulting dimensions do not match. Also, $A(x\odot y)\ne \mathit{𝐴𝑥}\odot \mathit{𝐴𝑦}$)

NOT Commutative:

$\mathit{𝐴𝑥}\ne \mathit{𝑥𝐴}$

$\mathit{𝐴𝐵}\ne \mathit{𝐵𝐴}$