Nonetheless, there are some useful inequalities relating the two, especially in the case of componentwise relative perturbations. It is shown that there can be no simple relationship between distance to singularity and condition number, because the condition number can be computed in polynomial time and the distance to singularity is NP-complete. One may again ask about distance to singularity, condition numbers, and complexity in this new context. More recent sensitivity analysis extends this analysis to perturbations of different maximum sizes in each entry of A. For the particular scenario under consideration, i.e., solution of PDEs, the coefficient matrix is rarely singular. If the coefficient matrix is singular, the matrix is not invertible. Using the algorithm of Golub and Kahan and its descendants, these quantities may be computed in $O(n^3 )$ operations. A matrix with a condition number equal to infinity is known as a singular matrix. As is well known, these two quantities, the minimum distance to singularity and the condition number, are essentially reciprocals. It also computes the condition number, or the sensitivity of $A^ $ to perturbations in A, where sensitivity is also measured with the two-norm.

First, it measures the distance from A to the nearest singular matrix, measuring distance with the two-norm.

The singular value decomposition of a square matrix A answers two questions.

Singular matrix