Condition Number

The condition number of an invertible matrix A is the product of the 2-norm of A and the 2-norm of A-1. This number measures the sensitivity of some solutions of linear equations Ax = b to perturbations in the entries of A and b. The matrix with condition number 1 is ``perfectly conditioned.''

$\blacktriangleright$ Matrices + Condition Number

$\left[\vphantom{ \begin{array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{array} }\right.$$\begin{array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{array}$$\left.\vphantom{ \begin{array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{array} }\right]$, condition number: 1

$\left[\vphantom{ \begin{array}{rrr} 5 & -5 & -3 \\ -3 & 0 & 5 \\ 1 & 5 & 4 \end{array} }\right.$$\begin{array}{rrr} 5 & -5 & -3 \\ -3 & 0 & 5 \\ 1 & 5 & 4 \end{array}$$\left.\vphantom{ \begin{array}{rrr} 5 & -5 & -3 \\ -3 & 0 & 5 \\ 1 & 5 & 4 \end{array} }\right]$, condition number: 2.8249

$\left[\vphantom{ \begin{array}{rrrr} 2 & -1 & 0 & 0 \\ -1 & 2 & -1 & 0 \\ 0 & -1 & 2 & -1 \\ 0 & 0 & -1 & 2 \end{array} }\right.$$\begin{array}{rrrr} 2 & -1 & 0 & 0 \\ -1 & 2 & -1 & 0 \\ 0 & -1 & 2 & -1 \\ 0 & 0 & -1 & 2 \end{array}$$\left.\vphantom{ \begin{array}{rrrr} 2 & -1 & 0 & 0 \\ -1 & 2 & -1 & 0 \\ 0 & -1 & 2 & -1 \\ 0 & 0 & -1 & 2 \end{array} }\right]$, condition number: $\left(\vphantom{ \frac{5}{2}+\frac{1}{2}\sqrt{5}}\right.$${\frac{{5}}{{2}}}$ + ${\frac{{1}}{{2}}}$$\sqrt{{5}}$$\left.\vphantom{ \frac{5}{2}+\frac{1}{2}\sqrt{5}}\right)$$\left(\vphantom{ \frac{3}{2}+\frac{1}{2}\sqrt{5}}\right.$${\frac{{3}}{{2}}}$ + ${\frac{{1}}{{2}}}$$\sqrt{{5}}$$\left.\vphantom{ \frac{3}{2}+\frac{1}{2}\sqrt{5}}\right)$ = 9.47214

$\left[\vphantom{ \begin{array}{cccc} 1 & \frac{1}{2} & \frac{1}{3} & \frac{1... ... \frac{1}{4} & \frac{1}{5} & \frac{1}{6} & \frac{1}{7} \end{array} }\right.$$\begin{array}{cccc} 1 & \frac{1}{2} & \frac{1}{3} & \frac{1}{4} \\ \frac{1}... ...{1}{6} \\ \frac{1}{4} & \frac{1}{5} & \frac{1}{6} & \frac{1}{7} \end{array}$$\left.\vphantom{ \begin{array}{cccc} 1 & \frac{1}{2} & \frac{1}{3} & \frac{1... ... \frac{1}{4} & \frac{1}{5} & \frac{1}{6} & \frac{1}{7} \end{array} }\right]$, condition number: 15514

$\left[\vphantom{ \begin{array}{cc} 1 & 1 \\ 1 & 1.00001 \end{array} }\right.$$\begin{array}{cc} 1 & 1 \\ 1 & 1.00001 \end{array}$$\left.\vphantom{ \begin{array}{cc} 1 & 1 \\ 1 & 1.00001 \end{array} }\right]$, condition number: 400002

These final two matrices are extremely ill conditioned. Small changes in some entries of A or b may result in large changes in the solution to linear equations of the form Ax = b in these two cases.