Numeric computation¶
Poor conditioning for matrix Inversion¶
Lets define the following function:
\[\begin{split}
f(x) = A^{-1}x \\
A \in R^{n \times n}
\end{split}\]
A is square thus it has eigenvalue decomposition
The conditioning number of the matrix is the magnitude of the largest and smallest eigenvalue
\[
\max_{i,j} |\frac{\lambda_i}{\lambda_j}|
\]
If the conditioning number is large, the matrix is sensitive to error in input. Poor conditioned matrices multiply pre-existing errors, and errors will be compounded further by numerical errors in inversion.