# kevop

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1. ## Changing roles in linear system for stability

Well I am not interested in the solution which is trivial to obtain but in the quality of future solutions. Let me put it this way: There are a total of H locations at which observations can be made.   Then if I have initially N locations in A (N >=4) at a given time then which of location from set H that are not in A will be most optimal for addition to A at N+1. That is when you use condition numbers to test the quality of possible future solutions for addition into A.   So the point is to test the quality of the future solution not using location in A but using the observation in B. By checking which one observation from H can improve the quality of future solution without ever solving the least square itself.    However checking condition number of B would be incorrect as it is not involved in inversion when finding X.  So there must a reason to relate condition number of A and B together for this linear system.
2. ## Changing roles in linear system for stability

That doesn't seem right. If you can multiply A times X, the number of columns in A has to match the number of rows in X. The transpose can take care of such issues. They are not important but if you wish it can be rewritten as:   A = N x 4 X = 4 x M B = N x M
3. ## Changing roles in linear system for stability

I want to test the stability of system using a non-invertible matrix B of the system instead of A, when X is unknown and A, B are both known. I am not interested in the solution directly but want to reduce error in the resulting solution by reducing condition number of B instead of A. All of them are matrices: X = 4 x M,  A = 4 x N, B = N x M.
4. ## Changing roles in linear system for stability

I have a linear system like: AX = B Normally to measure how stable the solution will be for such a system condition numbers are used. With X being the only unknown the condition number will be checked for A normally. However in this case A only defines the locations at which observations in B are made.   So checking the condition number and reducing it over multiple observations of A does not benefit me much as observed B at those locations might still be contributing to errors significantly in estimated X.   Instead if I check the condition number of B and try to reduce it over multiple observations by selecting A that reduces condition number of B I can more reliably estimate X.   But the issue is to justify the use of evaluating condition number for B which is not involved in inversion in the above linear system. Otherwise it seems incorrect to evaluate the condition number of B instead of A.   Normally condition numbers are defined to be the measure of relative error in X divided by the relative error in B etc.    It shows they are related so there must be a way to justify the use of minimizing condition number of B instead of A in this system and I want to know of some opinion or suggestions for this.