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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,

    $\displaystyle {\mathbf 0}_{2 \times 2} = \begin{bmatrix}0 & 0 \\ 0 & 0 \end{bma... ... {\mathbf 0}_{2 \times 3} = \begin{bmatrix}0 & 0 & 0\\ 0 & 0 & 0 \end{bmatrix}.$

  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}.$

next up previous contents
Next: Operations on Matrices Up: Definition of a Matrix Previous: Definition of a Matrix   Contents
A K Lal 2007-09-12