Condition Number

For a linear system the condition number, describes how small changes to will affect changes in . This essential tells us the sensitivity of the system to errors in .

To find the we start with the Relative Error Analysis formula:

Now assume that is Invertible, then we have:

Now we can transform equation using equation like so:

Next we use the operator norm definition and rearrange it to get we can then use this to further simplify .

This is exactly what we wanted:

So which we can then write as where Max & Min Stretch are the maximum and minimum stretch of .

The condition number has some important properties. Which can be used to quickly compute the condition number.