Linear Algebra Technique for Identifying Impactful Elements in a System

[mhdella]

Let’s say that we have a constant matrix A which is the coefficients matrix and column vector U of control variable as well as column vector X of state variables:
X=A*U
The question is: What is the proper technique in Linear Algebra that I should do to know which element in U has the most impact on the corresponding perturbed element in X.
On other words, there is an element in X has been perturbed and I would like to correct it by adjusting a few (as less as I can) elements in U.
I know the maximum element in the corresponding row of A which is multiplied by U column vector would have the most effect and by that I will know the corresponding element in U, but I am searching about a formal linear algebra technique to deal with this not algorithmic or programming procedure

 

[Bacle2]

Just an idea:

Is the matrix A diagonalizable? If so, maybe the diagonal form would make it
clearer .

 

[sunjin09]

If A is invertible, then the adjustment is unique, if A is rank deficient, the adjustment can be made minimal in L2 norm if you use pseudoinverse, if you want minimal L1 norm adjustment, you go with nonlinear optimization.

 

[mhdella]

i appreciate it. Thanx

 

[mhdella]

How can I do that by using singular value decomposition?