In a numerical analysis with a vector involvement, norms are essential to predict the various errors involved in the numerical analysis. A norm is a function in a vector space V. If A is an n*n matrix, then its norm is a real number. A norm satisfies the following properties:

  1. $   A   \geq 0$ for any square matrix $A$
  2. $   A   =0$; for null matrix
  3. $   kA   = k     A   $; $k$ is any scalar
  4. $   A+B   \leq   A   +   B   $
  5. $   AB   \leq   A   \quad   B   $

  6. Types of norm:

    1. 1- Norm or column norm: This norm is a maximum absolute sum of column and is defined as: \(\|A\|_{1}=\max _{1 \leq j \leq n}\left(\sum_{i=1}^{n}\left|a_{i j}\right|\right)\)

    2. Infinity norm: This norm is a maximum absolute sum of row and is defined as: \(\|A\|_{\infty}=\max _{1 \leq i \leq n}\left(\sum_{j=1}^{n}\left|a_{i j}\right|\right)\)

    3. 2- Norm or Eucledian norm: This norm is computed by taking square root of sum of squares of all the entities and it is defined as: \(\|A\|_{E}=\sqrt{\sum_{i=1}^{n} \sum_{j=1}^{n}\left(a_{i j}\right)^{2}}\)

    Consider a matrix A and its various norms are computed as: \(A=\left[\begin{array}{rrr} 5 & -4 & 2 \\ -1 & 3 & 2 \\ 1 & -2 & 0 \end{array}\right]\) \(\begin{aligned} \|A\|_{1} &=\max (5+1+1,4+3+2,2+2+0) \\ &=\max (7,9,4) \\ &=9 \end{aligned}\) \(\begin{aligned} \|A\|_{\infty} &=\max (5+4+2,1+3+2,1+2+0) \\ &=\max (11,6,3) \\ &=11 \end{aligned}\) \(\begin{aligned} \|A\|_{E} &=\sqrt{25+16+4+1+4+9+4+1+0} \\ &=\sqrt{64} \\ &=8 \end{aligned}\)

    Condition number:

    In the numerical analysis, the condition number of a matrix is very important since it represents how much change reflects in the output with a minor change in the input. If a condition number of a matrix is small, it is well conditioned problem and that can be handled efficiently and accurately while if the condition number is large, the problem is ill-conditioned and cannot be handled accurately. In that case, some preconditioners can be used. For a singular matrix, condition number is infinite since its determinant is zero and inverse is not possible. The condition number of an invertible matrix A is defined as: \(\kappa(A)=||A|| \quad\left||A^{-1}\right||\) The value of the condition number of a matrix is always greater than or equal to one. Also, it is noted that when we calculate the condition number of a matrix, same type of norm is considered both for the matrix and its inverse.

    \(A=\left[\begin{array}{rr} 2 & 3 \\ 1 & -1 \end{array}\right]\)\(\begin{gathered} \|A\|_{1}=\max (2+1,3+1)=4 \\ A^{-1}=\frac{1}{-2-3}\left[\begin{array}{rr} -1 & -3 \\ -1 & 2 \end{array}\right]=\left[\begin{array}{cc} \frac{1}{5} & \frac{3}{5} \\ \frac{1}{5} & \frac{-2}{5} \end{array}\right] \\ \left\|A^{-1}\right\|_{1}=\max \left(\frac{1}{5}+\frac{1}{5}, \frac{3}{5}+\frac{2}{5}\right)=1 \end{gathered}\)\(\text { Therefore } \kappa_{1}(A)=\|A\|_{1}\left\|A^{-1}\right\|_{1}=4 \times 1=4\)

    The convergence of an iterative process depends on the condition number of a matrix. If the square matrix of the linear algebraic equation $A x=b$ is singular, then this system does not have a solution. On the other hand, when the matrix is non-singular, it is the condition number of a matrix that decides about the convergence of the approximate solution obtained in the iteration process.