Matrix Condition Number

The condition number for matrix inversion with respect to a matrix norm $\lVert\cdot\rVert$ of a square matrix is defined by

$\displaystyle \kappa(A) = \Vert A \Vert \Vert A^{-1} \Vert\,,$

is non-singular; and $\kappa(A) = +\infty$ if

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 for a $5 \times 5$ Hilbert matrix) are said to be ill-conditioned.

If $\kappa(A)$ is the condition number of , then $\kappa(A)$ measures a sort of inverse distance from to the set of singular matrices, normalized by $\lVert A\rVert$ . Precisely, if is invertible, and $\lVert B - A\rVert < \lVert A^{-1}\rVert ^{-1}$ , then must also be invertible. On the other hand, in the case of the -norm, there always exists a singular matrix such that $\lVert B-A\rVert _2 = \lVert A^{-1}\rVert _2^{-1}$ (so the distance estimate is sharp).

Bibliography

1: Golub and Van Loan. Matrix Computations, 3rd edition. Johns Hopkins University Press, 1996.

"matrix condition number" is owned by stevecheng. [ full author list (2) | owner history (1) ]

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Other names:

matrix condition number, condition number

Also defines:

ill-conditioned, well-conditioned

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Cross-references: estimate, invertible, distance, sort, Hilbert matrix, near, operations, represents, linear system, matrix, measure, singular, non-singular, square matrix, matrix norm, matrix inversion
There are 6 references to this entry.

This is version 7 of matrix condition number, born on 2002-09-28, modified 2006-10-07.
Object id is 3480, canonical name is MatrixConditionNumber.
Accessed 43684 times total.

Classification:

AMS MSC:	65F35 (Numerical analysis :: Numerical linear algebra :: Matrix norms, conditioning, scaling)
	15A12 (Linear and multilinear algebra; matrix theory :: Conditioning of matrices)

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