Matrix Multiplication
Basic notations
Vector
Vector is usually referred as a column vector, which is a $k \times 1$ matrix. An example of $3 \times 1$ vector is as follow
Row vector, on the other hand, is the transpose of a column vector. The transpose of $\mathbf{v}$ is a $1 \times 3$ row vector.
Matrix
Matrix is denoted by $\textsf{#row} \times \textsf{#column}$. A $4 \times 3$ matrix looks like
$\mathbf{X} = \begin{pmatrix} 1 & 2 & 3\\ 1 & 2 & 3\\ 1 & 2 & 3 \\ 1 & 2 & 3 \\ \end{pmatrix}$Basic Arithmetic operations
Matrix product
Simply put, a $a \times b$ matrix multiplies a $b \times c$ matrix equals to a $a \times c$ matrix. If the number of columns of first matrix is not the same as the number of rows of the second matrix, these two matrices can not multiply each other.