2 Definitions

An array of numbers, rectangular in shape, is called a matrix . The first matrix below has 3 rows and 2 columns and is said to be a ‘3 by 2’ matrix (written $3\times 2$ ). The second matrix is a ‘2 by 4’ matrix (written $2\times 4$ ).

$\left[\begin{array}{cc}\hfill 1& \hfill 4\\ \hfill -2& \hfill 3\\ \hfill 2& \hfill 1\end{array}\right]\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill & \hfill 4\hfill \\ \hfill 5\hfill & \hfill 6\hfill & \hfill 7\hfill & \hfill 9\hfill \end{array}\right]$

The general $3\times 3$ matrix can be written

$A=\left[\begin{array}{ccc}{a}_{11}\hfill & a_{12}\hfill & a_{13}\hfill \\ a_{21}\hfill & a_{22}\hfill & a_{23}\hfill \\ a_{31}\hfill & a_{32}\hfill & a_{33}\hfill \end{array}\right]$

where $a_{ij}$ denotes the element in row $i$ , column $j$ .

For example in the matrix:

$A=\left[\begin{array}{ccc}0\hfill & -1\hfill & -3\hfill \\ 0\hfill & 6\hfill & -12\hfill \\ 5\hfill & 7\hfill & 123\hfill \end{array}\right]$

$a_{11}=0,a_{12}=-1,a_{13}=-3,\dots a_{22}=6,\dots a_{32}=7,a_{33}=123$

A matrix with only one column is called a column vector (or column matrix ).

For example, $\left[\begin{array}{c}\hfill x_1\hfill \\ \hfill x_2\hfill \\ \hfill x_3\hfill \end{array}\right]$ and $\left[\begin{array}{c}\hfill 3\hfill \\ \hfill 4\hfill \\ \hfill 5\hfill \end{array}\right]$ are both $3\times 1$ column vectors.

A matrix with only one row is called a row vector (or row matrix ). For example $\left[2,-3,8,9\right]$ is a $1\times 4$ row vector. Often the entries in a row vector are separated by commas for clarity.

2.1 Square matrices

When the number of rows is the same as the number of columns, i.e. $m=n$ , the matrix is said to be square and of order $n$ (or $m$ ).

• In an $n\times n$ square matrix $A$ , the leading diagonal (or principal diagonal ) is the ‘north-west to south-east’ collection of elements $a_{11},a_{22},\dots ,a_{nn}.$ The sum of the elements in the leading diagonal of $A$ is called the trace of the matrix, denoted by tr $\left(A\right)$ .

  $A=\left[\begin{array}{cccc}\hfill a_{11}\hfill & \hfill a_{12}\hfill & \hfill \dots \hfill & \hfill a_{1n}\hfill \\ \hfill a_{21}\hfill & \hfill a_{22}\hfill & \hfill \dots \hfill & \hfill a_{2n}\hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill \\ \hfill a_{n1}\hfill & \hfill a_{n2}\hfill & \hfill \dots \hfill & \hfill a_{nn}\hfill \end{array}\right]\text{tr}\left(A\right)=a_{11}+a_{22}+\cdots +a_{nn}$

• A square matrix in which all the elements below the leading diagonal are zero is called an upper triangular matrix , often denoted by $U$ .

  $U=\left[\begin{array}{ccccc}\hfill u_{11}\hfill & \hfill u_{12}\hfill & \hfill \dots \hfill & \hfill \dots \hfill & \hfill u_{1n}\hfill \\ \hfill 0\hfill & \hfill u_{22}\hfill & \hfill \dots \hfill & \hfill \dots \hfill & \hfill u_{2n}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \dots \hfill & \hfill ⋮\hfill & \hfill ⋮\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \dots \hfill & \hfill 0\hfill & \hfill u_{nn}\hfill \end{array}\right]u_{ij}=0\text{when}i>j$

• A square matrix in which all the elements above the leading diagonal are zero is called a lower triangular matrix , often denoted by $L$ .

  $L=\left[\begin{array}{ccccc}\hfill l_{11}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \dots \hfill & \hfill 0\hfill \\ \hfill l_{21}\hfill & \hfill l_{22}\hfill & \hfill 0\hfill & \hfill \dots \hfill & \hfill 0\hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \dots \hfill & \hfill \dots \hfill & \hfill 0\hfill \\ \hfill l_{n1}\hfill & \hfill l_{n2}\hfill & \hfill ⋮\hfill & \hfill \dots \hfill & \hfill l_{nn}\hfill \end{array}\right]l_{ij}=0\text{when}i<j$

• A square matrix where all the non-zero elements are along the leading diagonal is called a diagonal matrix, often denoted by $D$ .

  $D=\left[\begin{array}{ccccc}\hfill d_{11}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \dots \hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill d_{22}\hfill & \hfill 0\hfill & \hfill \dots \hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \dots \hfill & \hfill \dots \hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \dots \hfill & \hfill d_{nn}\hfill \end{array}\right]d_{ij}=0\text{when}i\ne j$

2.2 Some examples of matrices and their classification

$A=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill \\ \hfill 4\hfill & \hfill 5\hfill & \hfill 6\hfill \end{array}\right]$ is $2\times 3$ . It is not square.

$B=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 2\hfill \\ \hfill 3\hfill & \hfill 4\hfill \end{array}\right]$ is $2\times 2$ . It is square.

Also, tr $\left(A\right)$ does not exist, and tr $\left(B\right)=1+4=5$ .

$C=\left[\begin{array}{ccc}\hfill 1& \hfill 2& \hfill 3\\ \hfill 0& \hfill -2& \hfill -5\\ \hfill 0& \hfill 0& \hfill 1\end{array}\right]$ and $D=\left[\begin{array}{ccc}\hfill 4& \hfill 0& \hfill 3\\ \hfill 0& \hfill -2& \hfill 5\\ \hfill 0& \hfill 0& \hfill 1\end{array}\right]$ are both $3\times 3$ , square and upper triangular.

Also, tr $\left(C\right)=0$ and tr $\left(D\right)=3$ .

$E=\left[\begin{array}{ccc}\hfill 1& \hfill 0& \hfill 0\\ \hfill 2& \hfill -2& \hfill 0\\ \hfill 3& \hfill -5& \hfill 1\end{array}\right]$ and $F=\left[\begin{array}{ccc}\hfill -1& \hfill 0& \hfill 0\\ \hfill 1& \hfill 4& \hfill 0\\ \hfill 0& \hfill 1& \hfill 1\end{array}\right]$ are both $3\times 3$ , square and lower triangular.

Also, tr $\left(E\right)=0$ and tr $\left(F\right)=4$ .

$G=\left[\begin{array}{ccc}\hfill 1& \hfill 0& \hfill 0\\ \hfill 0& \hfill 2& \hfill 0\\ \hfill 0& \hfill 0& \hfill -3\end{array}\right]$ and $H=\left[\begin{array}{ccc}\hfill 4\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]$ are both $3\times 3$ , square and diagonal.

Also, tr $\left(G\right)=0$ and tr $\left(H\right)=6$ .

Task!

Classify the following matrices (and, where possible, find the trace):

$A=\left[\begin{array}{cc}1\hfill & 2\hfill \\ 3\hfill & 4\hfill \\ 5\hfill & 6\hfill \end{array}\right]B=\left[\begin{array}{cccc}1\hfill & 2\hfill & 3\hfill & 4\hfill \\ 5\hfill & 6\hfill & 7\hfill & 8\hfill \\ -1\hfill & -3\hfill & -2\hfill & -4\hfill \end{array}\right]C=\left[\begin{array}{cccc}1\hfill & 2\hfill & 3\hfill & 4\hfill \\ 5\hfill & 6\hfill & 7\hfill & 8\hfill \\ 9\hfill & 10\hfill & 11\hfill & 12\hfill \\ 13\hfill & 14\hfill & 15\hfill & 16\hfill \end{array}\right]$

Answer

Task!

Classify the following matrices:

$A=\left[\begin{array}{ccc}1\hfill & 1\hfill & 1\hfill \\ 1\hfill & 1\hfill & 1\hfill \\ 1\hfill & 1\hfill & 1\hfill \end{array}\right]B=\left[\begin{array}{ccc}1\hfill & 0\hfill & 0\hfill \\ 1\hfill & 1\hfill & 0\hfill \\ 1\hfill & 1\hfill & 1\hfill \end{array}\right]C=\left[\begin{array}{ccc}1\hfill & 1\hfill & 1\hfill \\ 0\hfill & 1\hfill & 1\hfill \\ 0\hfill & 0\hfill & 1\hfill \end{array}\right]D=\left[\begin{array}{ccc}1\hfill & 0\hfill & 0\hfill \\ 0\hfill & 1\hfill & 0\hfill \\ 0\hfill & 0\hfill & 1\hfill \end{array}\right]$

Answer

2.3 Equality of matrices

As we noted earlier, the terms in a matrix are called the elements of the matrix.

$\text{The elements of the matrix}A=\left[\begin{array}{cc}1\hfill & 2\hfill \\ -1\hfill & -4\hfill \end{array}\right]\text{are}1,2,-1,-4$

We say two matrices $A$ , $B$ are equal to each other only if $A$ and $B$ have the same number of rows and the same number of columns and if each element of $A$ is equal to the corresponding element of $B$ . When this is the case we write $A=B$ . For example if the following two matrices are equal:

$A=\left[\begin{array}{cc}1\hfill & \alpha \hfill \\ -1\hfill & -\beta \hfill \end{array}\right]B=\left[\begin{array}{cc}1\hfill & 2\hfill \\ -1\hfill & -4\hfill \end{array}\right]$

then we can conclude that $\alpha =2$ and $\beta =4$ .

2.4 The unit matrix

The unit matrix or the identity matrix , denoted by $I_n$ (or, often, simply $I$ ), is the diagonal matrix of order $n$ in which all diagonal elements are 1.

Hence, for example, $I_2=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$ and $I_3=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$ .

2.5 The zero matrix

The zero matrix or null matrix is the matrix all of whose elements are zero. There is a zero matrix for every size. For example the $2\times 3$ and $2\times 2$ cases are:

$\left[\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right],\left[\begin{array}{cc}\hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right].$

Zero matrices, of whatever size, are denoted by $\underset{̲}{0}$ .

2.6 The transpose of a matrix

The transpose of a matrix $A$ is a matrix where the rows of $A$ become the columns of the new matrix and the columns of $A$ become its rows. For example

$A=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill \\ \hfill 4\hfill & \hfill 5\hfill & \hfill 6\hfill \end{array}\right]\text{becomes}\left[\begin{array}{cc}\hfill 1\hfill & \hfill 4\hfill \\ \hfill 2\hfill & \hfill 5\hfill \\ \hfill 3\hfill & \hfill 6\hfill \end{array}\right]$

The resulting matrix is called the transposed matrix of $A$ and denoted $A^T$ . In the previous example it is clear that $A^T$ is not equal to $A$ since the matrices are of different sizes. If $A$ is square $n\times n$ then $A^T$ will also be $n\times n$ .

Example 1

Find the transpose of the matrix $B=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill \\ \hfill 4\hfill & \hfill 5\hfill & \hfill 6\hfill \\ \hfill 7\hfill & \hfill 8\hfill & \hfill 9\hfill \end{array}\right]$

Solution

Interchanging rows with columns we find

$B^T=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 4\hfill & \hfill 7\hfill \\ \hfill 2\hfill & \hfill 5\hfill & \hfill 8\hfill \\ \hfill 3\hfill & \hfill 6\hfill & \hfill 9\hfill \end{array}\right]$

Both matrices are $3\times 3$ but $B$ and $B^T$ are clearly different.

When the transpose of a matrix is equal to the original matrix i.e. $A^T=A$ , then we say that the matrix $A$ is symmetric . (This is because it has symmetry about the leading diagonal.)

In Example 1 $B$ is not symmetric.

Example 2

Show that the matrix $C=\left[\begin{array}{ccc}\hfill 1& \hfill -2& \hfill 3\\ \hfill -2& \hfill 4& \hfill -5\\ \hfill 3& \hfill -5& \hfill 6\end{array}\right]$ is symmetric.

Solution

Taking the transpose of $C$ :

$C^T=\left[\begin{array}{ccc}\hfill 1& \hfill -2& \hfill 3\\ \hfill -2& \hfill 4& \hfill -5\\ \hfill 3& \hfill -5& \hfill 6\end{array}\right]$ .

Clearly $C^T=C$ and so $C$ is a symmetric matrix. Notice how the leading diagonal acts as a “mirror”; for example $c_{12}=-2$ and $c_{21}=-2$ . In general $c_{ij}=c_{ji}$ for a symmetric matrix.

Task!

Find the transpose of each of the following matrices. Which are symmetric?

$A=\left[\begin{array}{cc}1\hfill & 2\hfill \\ 3\hfill & 4\hfill \end{array}\right],B=\left[\begin{array}{cc}1\hfill & 1\hfill \\ -1\hfill & 1\hfill \end{array}\right]C=\left[\begin{array}{cc}1\hfill & 1\hfill \\ 1\hfill & 0\hfill \end{array}\right]$

$D=\left[\begin{array}{cc}1\hfill & 2\hfill \\ 4\hfill & 5\hfill \\ 7\hfill & 8\hfill \end{array}\right]E=\left[\begin{array}{cc}1\hfill & 0\hfill \\ 0\hfill & 1\hfill \end{array}\right]$

Answer