matrix condition number

1 Matrix Condition Number

The condition numberMathworldPlanetmath for matrix inversionMathworldPlanetmath with respect to a matrix normMathworldPlanetmath of a square matrixMathworldPlanetmath A is defined by


if A is non-singular; and κ(A)=+ if A is singular.

The condition number is a measure of stability or sensitivity of a matrix (or the linear system it represents) to numerical operations. In other words, we may not be able to trust the results of computations on an ill-conditioned matrix.

Matrices with condition numbers near 1 are said to be well-conditioned. Matrices with condition numbers much greater than one (such as around 105 for a 5×5 Hilbert matrixMathworldPlanetmath) are said to be ill-conditioned.

If κ(A) is the condition number of A, then κ(A) measures a sort of inverse distance from A to the set of singular matrices, normalized by A. Precisely, if A is invertible, and B-A<A-1-1, then B must also be invertible. On the other hand, in the case of the 2-norm, there always exists a singular matrix B such that B-A2=A-12-1 (so the distance estimate is sharp).


  • 1 Golub and Van Loan. Matrix Computations, 3rd edition. Johns Hopkins University Press, 1996.
Title matrix condition number
Canonical name MatrixConditionNumber
Date of creation 2013-03-22 13:04:17
Last modified on 2013-03-22 13:04:17
Owner stevecheng (10074)
Last modified by stevecheng (10074)
Numerical id 10
Author stevecheng (10074)
Entry type Definition
Classification msc 15A12
Classification msc 65F35
Synonym matrix condition number
Synonym condition number
Related topic PropertyOfMatrixConditionNumber
Defines ill-conditioned
Defines well-conditioned