Hearing Perturbation Theory

In numerical linear algebra, we create ways for a computer to solve a linear system of equations A\vec{x} = \vec{b}. In doing so, we analyze how efficiently and accurately we can find the solution \vec{x}.

Perturbation theory concerns how much error we incur in the solution \vec{x} when we perturb (spoil) the data A and \vec{b}. A classic statement tells us that the amount of error depends on the condition number of the matrix A.

I will define and prove the statement, and help you understand it by “hearing” it.

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