# What is a big condition number for a matrix?

The condition number of a matrix is a measure of how close a matrix is to being singular.

But, what is considered a big condition number?

• This cannot be determined rigorously. Did you find any suggestion what could be considered to be "big" ? I remember values like $1000$, but there is no sharp border-line. Mar 3, 2019 at 16:39
• $10^5$ and larger Jan 10, 2020 at 19:05

You can view a solver as a black box which maps the input (in your case, the right-hand side vector) into the output (in your case the solution). The condition number connects the relative error of the input to the relative error of the output. Ideally, we have a relationship of the form $$\text{relative error on the output} \leq \text{condition number} \times \text{relative error on the input}.$$ This assumes well written code with no rounding errors made during the computation. While this is nearly impossible to achieve in practice, relations of this type set clear limits for what can be achieved. If we need a relative error of $$10^{-6}$$ on the output side and we have a relative error of $$10^{-16}$$ on the input side, then a condition number of $$10^{10}$$ is the largest value we can tolerate. Anything larger is too large, anything smaller is just a bonus.