Types of matrices

Matrices like any other mathematical tool make our life easier! Of course, they are not as simplistic as multiplication or division operations which help us every day in simple concepts such as finding out how much 2 kilograms of apple would cost, but they are still present around us in a myriad of other, more sophisticated processes. Every time you surf the web some data is encrypted and for that, we need... you guessed it – matrices! Economist makes a report or data is analyzed, once again matrices save the day. Imagine a manager in a grocery store who wants to compare clients according to their purchases. What do they do? They represent your receipt as a matrix where each element represents the number of a particular commodity purchased and afterward those matrices are compared! Now that you know how broadly applicable matrices are, we hope that you are motivated to start learning about different types of matrices and notions related to them!

Basic matrices

A square matrix $n\times n$ is a matrix with the same $n$ number of rows and columns. For example, this is a square matrix $3\times3$:

$\begin{pmatrix} 1 & 0 & 3 \\ 2 & -1 & 4 \\ 11 & 7 & 11 \end{pmatrix}$A zero matrix is a matrix with every element equal to $0$:

$\begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}$

Diagonal matrices

The first non-zero element in each row is called the leading entry. In the example below, the leading entries are colored red:

$\begin{pmatrix} {\color{red}1} & 2 & 6 & 3 \\ 0 & 0 & 0 & {\color{red}7} \\ 0 & 0 & 0 & 0 \\ 0 & {\color{red}1} & -1 & 0 \end{pmatrix}$

A diagonal matrix is a square matrix where all leading entries are located at the diagonal from the top left to the bottom right corner (and have, therefore, indices $a_{1,1}, a_{2,2}, a_{3,3}\dots a_{n,n}$), and all other elements are zeroes. This diagonal is known as the main diagonal of the matrix, in the example below, the main diagonal is colored red:$\begin{pmatrix} \color{red}{1} & 0 & 0 \\ 0 & \color{red}{3} & 0 \\ 0 & 0 & \color{red}{7} \end{pmatrix}$

A scalar matrix is a diagonal matrix whose main diagonal elements all contain the same scalar number value: $\begin{pmatrix} \color{red}{7} & 0 & 0 \\ 0 & \color{red}{7} & 0 \\ 0 & 0 & \color{red}{7} \end{pmatrix}$

In non-square $n \times m$ matrices the main diagonal elements, too, have indices $a_{1,1}, a_{2,2}, a_{3,3}\dots a_{\min{(m,n)},\min{(m,n)}}$:

$\begin{pmatrix} {\color{red}9} & 0 & 0 \\ 0 & {\color{red}13} & 0 \\ 0 & 0 & {\color{red}-7} \end{pmatrix} \begin{pmatrix} {\color{red}1} & 0 & 0 & 0 \\ 0 & {\color{red}3} & 0 & 0 \\ 0 & 0 & {\color{red}5} & 0 \end{pmatrix} \begin{pmatrix} {\color{red}-1} & 0 & 0 \\ 0 & {\color{red}42} & 0 \\ 0 & 0 & {\color{red}8} \\ 0 & 0 & 0 \end{pmatrix}$

The diagonal from the bottom left to the top right corner of a square matrix, containing elements $a_{n,1}, a_{n-1,2}, a_{n-2,3}\dots a_{2,n-1},a_{1,n}$, is, therefore, called the secondary diagonal:

$\begin{pmatrix} 0 & 0 & {\color{red}6} \\ 0 & {\color{red}4} & 0 \\ {\color{red}-3} & 0 & 0 \end{pmatrix}$

An identity matrix (sometimes ambiguously called a unit matrix) of size $n$ is the $n\times n$ square matrix with elements equal to $1$ on the main diagonal and $0$ elsewhere. It is denoted by $I_{n}$, or simply by $I$, if the size of a matrix does not matter. For example, below is a $3\times3$ identity matrix.$I_{3} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$

The name of the identity matrix comes from the fact that multiplying by it would not change any matrix in any way, so the resulting matrix is identical to the original matrix. So, we can think of it as multiplying by $1$ if we draw parallels to multiplying numbers.

Triangular matrices

An upper triangular matrix is a square matrix where all elements below the main diagonal are equal to $0$: $a_{i,j}=0$, if $i>j$. All other elements can be arbitrary (including 0):

$\begin{pmatrix} 7 & 2 & 1& 0 \\ \color{red}{0} & 0 & 4 & 1 \\ \color{red}{0} & \color{red}{0} & 7 & 3 \\ \color{red}{0} & \color{red}{0} & \color{red} 0 & 6\end{pmatrix}$

A lower triangular matrix is a square matrix where all elements above the main diagonal are equal to $0$: $a_{i,j}=0$, if $i<j$. Analogously to the upper triangular matrix, all other elements can be arbitrary:$\begin{pmatrix} 7 & \color{red}{0} & \color{red}{0} & \color{red} 0\\ 3 & 7 & \color{red}{0} & \color{red} 0 \\ 1 & 2 & 0 & \color{red} 0 \\ 6 & 0 & 8 & 9\end{pmatrix}$

A symmetric matrix is a square matrix where $a_{i,j} = a_{j,i}$

$\begin{pmatrix} 7 & \color{red}{0} & \color{green}{1} \\ \color{red}{0} & 7 & \color{blue}{2} \\ \color{green}{1} & \color{blue}{2} & 2 \end{pmatrix}$

Note that square zero matrices and identity matrices are special cases of scalar matrices, scalar matrices are special cases of diagonal matrices and diagonal matrices are in turn special cases of lower triangular, upper triangular, and symmetric matrices.

Conclusion

Let's summarize! In this topic, we got to know different types of matrices, why they are useful and how can we apply them in practice, here are some of the main points:

• A square matrix is a matrix that has the same number of rows and columns and so has a dimension $n\times n$

• Zero matrices are matrices that contain only $0$s

• A leading entry is the first nonzero element of every row in a matrix

• A diagonal matrix has nonzero elements only on its diagonal, hence the name, while a matrix with nonzero elements on its secondary diagonal is called a secondary diagonal matrix

• An identity matrix is a square diagonal matrix with all its nonzero elements equal to 1, if we multiply a matrix by an identity matrix we will get the same matrix

• Triangular matrices are those matrices that have all their elements equal to $0$s below or above the main diagonal