How this calculator works
The condition number compares how much a matrix can stretch a vector with how much its inverse can. It bounds the relative error in a computed solution for a relative error in the input. A condition number of one is ideal and occurs for orthogonal matrices in the 2-norm. A singular matrix has an infinite condition number because no inverse exists. The 1-norm and infinity-norm use maximum column and row sums; the 2-norm uses singular values; the Frobenius and entrywise max norms are not induced operator norms but are useful practical measures.
Formula
kappa(A) = ||A|| ||A inverse||. In the 2-norm, kappa = sigma max / sigma min.
Worked example
The matrix with rows 1, 2 and 3, 4 has 1-norm 6, and its inverse has 1-norm 3.5, so the condition number is 21. Its 2-norm condition number is about 14.933. A diagonal matrix with diagonal 1, 2, 4 has 2-norm condition number 4.
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Frequently asked questions
What counts as a large condition number?
It depends on the precision you need. In double precision, a condition number near 10^12 leaves only a few trustworthy digits in a solved system; near 10^16 the result is essentially noise.
Why does a singular matrix have an infinite value?
The smallest singular value is zero, so the ratio of largest to smallest is infinite. Equivalently, the inverse does not exist and the error bound cannot be finite.
Can changing the norm change the value?
Yes. Different norms give different numbers, but they usually agree on the order of magnitude. The 2-norm is the most common choice in analysis.