In this small topic, I will show several traditional results regarding to the perturbation in matrix algebra. This comes from the study of the stability of solving system of linear equations in Numerical Analysis.
Let us consider the following matrix-form linear system 
. We split into three cases
- Perturbation on
If 
is a small perturbation on , then 
is a small perturbation on 
in the sense that the following system 
holds true. This together with the fact that 
gives 
. By some simple calculation, one has
Thus
- Perturbation on
.
If 
is a small perturbation on , then is a small perturbation on in the sense that the following system 
holds true. This together with the fact that gives 
. By some simple calculation, one has
Thus
which implies that
- Perturbation on and .
If is a small perturbation on and is a small perturbation on , then is a small perturbation on in the sense that the following system 
holds true. This together with the fact that gives 
. By some simple calculation, one has
which implies that
Thus,
In the literature, the expression 
is called the condition number of the matrix , denoted by 
. It is easy to see that 
. If is small then the system is called to be well-posed. Alternatively, in the case when 
, the system is ill-posed.
