M.1 Matrix Definitions

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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:

\[ E = \pi \]

\[ F = 6 \]