M.1 Matrix Definitions
M.1 Matrix Definitions Matrix

A matrix is a rectangular collection of numbers. Generally, matrices are denoted as bold capital letters. For example:
\[A = \begin{pmatrix} 1 & 5 & 4\\
2 & 5 & 3 \end{pmatrix}\]
A is a matrix with two rows and three columns. For that reason, it is called a 2 by 3 matrix. This is called the dimension of a matrix.
 Dimension

The dimension of a matrix is expressed as number of rows × number of columns. So,
\[B = \begin{pmatrix} 1 & 5 & 4 \\ 5 & 3 & 8 \\ 1 & 5 & 4 \\ 2 & 5 & 3 \end{pmatrix}\]
B is a 4 × 3 matrix. It is common to refer to elements in a matrix by subscripts, like so.
\[B = \begin{pmatrix} b_{1,1} & b_{1,2} & b_{1,3}\\ b_{2,1} & b_{2,2} & b_{2,3}\\ b_{3,1} & b_{3,2} & b_{3,3}\\ b_{4,1} & b_{4,2} & b_{4,3} \end{pmatrix}\]
With the row first and the column second. So in this case, \(b_{2,1} = 5\) and \(b_{1,3} =4\).
 Vector

A vector is a matrix with only one row (called a row vector) or only one column (called a column vector). For example:
\[C = \begin{pmatrix} 2 & 7 & 3 & 5 \end{pmatrix}\]
C is a 4 dimensional row vector.
\[D = \begin{pmatrix} 2 \\9 \\3 \\3\\ 6\\ \end{pmatrix}\]
D is a 5 dimensional column vector. An "ordinary" number can be thought of as a 1 × 1 matrix, also known as a scalar. Some examples of scalars are shown below:
\[ E = \pi \]
\[ F = 6 \]