Definition – Matrix. – Math-reference.net

Contents
1. Definition
2. Common Notation
3. Matrix Multiplication
4. Matrix Addition
5. Square Matrices, Leading Diagonal and Trace
6. Identity Matrix
7. Inverse Matrix
8. Matrices as linear transformations

Definition

Let $S$ be a set. Then, a matrix is a collection of elements of $S$ , denoted $s_{i,j}$ , arranged in the form,

$\begin{pmatrix} s_{1,1} & s_{1,2} & ... & s_{1,n} \\ s_{2,1} & s_{2,2} & ... & s_{2,n}\\ \vdots & \vdots & \ddots & \vdots \\ s_{m,1}& s_{m,2} & ... & s_{m,n} \end{pmatrix}$

Where $s_{i,j}$ denotes the element in the $i$ -th row, and the $j$ -th column, starting from the top left element and counting downwards for rows and to the right for columns. Note that the reason for using the dots is that they fill in for several elements.

Dimension of a Matrix

The dimension of a matrix $A$ , where $A$ has $m$ rows and $n$ columns, is defined to be $m \times n$ . Ie, the dimension of the matrix

$A = \begin{pmatrix} 11 & -105 & 49 \\64& 2 & -56 \end{pmatrix}$

is $2\times 3$ . We also say that $A$ is an $m\times n$ matrix as a synonym for $A$ having dimension $m\times n$ .

Common notation

It is convention to denote the elements of a matrix using small letters, and the matrices themselves by capital letters. Furthermore, whatever letter is used for the matrix, we use the lower case version of that letter to denote its elements.

If we wish to refer to a specific element in the matrix, say the matrix $A$ , then we may do so by specifiying the row and column in which the element is located. To do so, we use the format $a_{i,j}$ , as in the definition. For example, in the matrix

$A = \begin{pmatrix} 11 & -105 & 49 \\64& 2 & -56\\ \ -19 & 41 & 29 \end{pmatrix}$

the element $a_{2,3}$ is $-56$

Matrix Multiplication

The multiplication of two matrices, $A$ and $B$ , is written $AB$ , and is only defined when the number of columns of $A$ is equal to the number of rows of $B$ .

If $C=AB$ , where $A$ has dimension $m\times n$ and $B$ has dimension $n \times q$ , then the element $c_{i,j}$ of $C$ is given by the following formula,

$c_{i,j} = \sum\limits_{k = 1}^n a_{i,k} b_{k,j}$

and $C$ will have dimension $m \times q$ . Ie, the formula above is valid for $i = 1, ..., m$ and $j = 1, ..., q$ .

Matrix Addition and Subtraction

We define matrix addition similarly. Let $C = A\pm B$ . Then, the elements of $C$ are defined by the formula,

$c_{i,j} = a_{i,j} \pm b_{i,j}$

Thus, addition and subtraction between matrices is only defined if they have the same dimension.

Square Matrices, Leading Diagonal and Trace

A square matrix is defined as a matrix with dimension $n\times n$ . In other words, it is a matrix which has the same number of rows as it has columns.

The leading diagonal of a matrix $A$ is defined only for square matrices. Let $A$ be a square matrix with dimension $n\times n$ . Then, the leading diagonal of $A$ is the list of elements,

$\text{Leading Diagonal of A} = (c_{1,1}, c_{2,2}, ..., c_{n,n})$

That is, it is the set of elements where the row number is equal to the column number. For example, the leading diagonal of the matrix

$T = \begin{pmatrix} 11 & -105 & 49 \\64& 2 & -56\\ \ -19 & 41 & 29 \end{pmatrix}$

Is the ordered set of elements,

$(11, 2, 29)$

The trace of a matrix $A$ , denoted by $tr(A)$ , is only defined for square matrices. The trace is defined to be the sum of the elements in the leading diagonal. For matrix $T$ above, the trace is equal to $11 + 2 + 29 = 52$

The Identity Matrix

The identity matrix of dimension $n$ , denoted by $I_n$ is the $n\times n$ matrix such that for every element of the matrix $i_{x,y}$ ,

$i_{i,j} = \bigg \{ \begin{array}{cc} 1, & i=j \\ 0, & i\neq j\end{array}$

For all $i = 1, ..., n$ and $j = 1,..., n$ . An example of a $2\times 2$ matrix would be,

$I_2 = \left( \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right)$

The determinant of a matrix

Please note that the determinant section is still under review as I need time to derive the properties of determinants etc using this definition of a determinant.

Given a matrix $A$ , we can perform matrix multiplication on a set of vectors. This will give us a new matrix of vectors, which we may find the ‘volume’ of. The ratio $\frac{new shape}{old shape}$ of the new to old volumes is equal to the determinant of the matrix $A$ .

The determinant of $A$ is only defined if $A$ is a square matrix.