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For a given problem, A*X = B where A is a NxN matrix, X and B are Nx1 accordingly, I understand that a poor condition number (high condition number) implies that the matrix A is poorly invertible (nearly singular). Numerically, a small error in B results in a large error in X.

Now, specifically my problem is:

- I know the true value of the matrix X

- When this system is solved using: X = A^-1*B, it gives an understandably poor answer.

- When I use a nonlinear system solver (a solver that solves a system of nonlinear equations), I get a resulting matrix X* (the optimal X found by the algorithm) where the Rows and Columns of X* sum up to the correct values (if you summed row 1 of X*, it's the same sum as row 1 of Xtrue etc.), however the actual values of each cell have huge errors.

Now, is there an underlying mathematical reason for this? Numerous nonlinear solvers were used all with the same result (including algorithms such as simulated annealing).

The only thing I can personally come up with is:

If matrix A had a poor condition number, then it's almost like the system has multiple solutions. In which case, it's analogous to a system of equations where each equation was a multiple of another one. As long as the sum of the variables is satisfied, it's a valid solution.

This is only a hunch, and I'd like some affirmation or any other explanations.

Thanks.

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# Condition # and Linear System Solution

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