Special Matrices

DEFINITION 1.1.5  

1. A matrix in which each entry is zero is called a zero-matrix, denoted by ${\mathbf 0}.$ For example,
   $${\mathbf 0}_{2 \times 2} = \begin{bmatrix}0 & 0 \\ 0 & 0 \end{bma... ... {\mathbf 0}_{2 \times 3} = \begin{bmatrix}0 & 0 & 0\\ 0 & 0 & 0 \end{bmatrix}.$$

2. A matrix for which the number of rows equals the number of columns, is called a square matrix. So, if $A$ is an $n \times n$ matrix then $A$ is said to have order $n$ .

3. In a square matrix, $A=[a_{ij}],$ of order $n$ , the entries $a_{11}, a_{22}, \ldots, a_{nn}$ are called the diagonal entries and form the principal diagonal of $A.$

4. A square matrix $A=[a_{ij}]$ is said to be a diagonal matrix if $\; a_{ij} = 0$ for $i \neq j.$ In other words, the non-zero entries appear only on the principal diagonal. For example, the zero matrix ${{\mathbf 0}}_n$ and $\begin{bmatrix}4&0\\ 0&1 \end{bmatrix}$ are a few diagonal matrices.

   A diagonal matrix $D$ of order $n$ with the diagonal entries $d_1, d_2, \ldots, d_n$ is denoted by $D= {\mbox{diag}}(d_1, \ldots, d_n).$

   If $d_i=d$ for all $i = 1, 2, \ldots, n$ then the diagonal matrix $D$ is called a scalar matrix.

5. A diagonal matrix $A$ of order $n$ is called an IDENTITY MATRIX if $d_i = 1$ for all $i = 1, 2, \ldots, n$ . This matrix is denoted by $I_n$ .

   For example, $I_2 = \begin{bmatrix}1&0\\ 0&1 \end{bmatrix}$ and $I_3 = \begin{bmatrix}1&0&0\\ 0&1&0 \\ 0&0&1 \end{bmatrix}.$

   The subscript $n$ is suppressed in case the order is clear from the context or if no confusion arises.

6. A square matrix $A=[a_{ij}]$ is said to be an upper triangular matrix if $a_{ij} = 0$ for $i > j.$

   A square matrix $A=[a_{ij}]$ is said to be a lower triangular matrix if $a_{ij} = 0$ for $i < j.$

   A square matrix $A$ is said to be triangular if it is an upper or a lower triangular matrix.

   For example $\begin{bmatrix}2 & 1 & 4\\ 0&3&-1\\ 0&0&-2 \end{bmatrix}$ is an upper triangular matrix. An upper triangular matrix will be represented by $\begin{bmatrix}a_{11} & a_{12} & \cdots & a_{1n} \\ 0 & a_{22} & \cdots & a_{... ...\\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & a_{nn} \end{bmatrix}.$