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I've got a general question. For the sum of the matrices $\boldsymbol C = \boldsymbol A + \boldsymbol B$, under which conditions would one say that the approxmation $\boldsymbol C \approx \boldsymbol A $ is good and valid. What are the criteria to justify this approximation?

Thank you!

bonanza
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1 Answers1

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That really depends on what you plan to do with the matrices. Suppose we want to solve the linear systems $Ax=d$ and $Cx=d$, for $d=[4~~ 7.999]^T$. If $$A=\left[\begin{smallmatrix} 1&2\\2&3.999\end{smallmatrix}\right], B=\left[\begin{smallmatrix} 0.001&0\\0&0\end{smallmatrix}\right]$$

we might think that $A,C$ are close, but the solution for $A$ is $[2~~ 1]^T$ while the solution for $C$ is $[-0.667~~ 2.334]^T$.

If this happens we call $A$ ill-conditioned. See here for some examples and discussion of this.

vadim123
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  • Thanks for your answer! In fact your right and I have to solve a set of linear equations: $\boldsymbol C \boldsymbol x = \boldsymbol b$. Furthermore A is symmetric and positive semidefinite. And I want to know whether I can save computational time by the approximation above. Analytically I can compute the eigenvalues and eigenvectors of $A,B,C$, does this help to make a good statement? For the solution of the linear system the eigenvalues and -vectors of $C$ are quite important, right? Would make sense to say that if the eigenvalues of $B$ are small this approx. is valid? – bonanza Jul 08 '13 at 07:20
  • Eigenvalues don't really help you; in my example $B$ has small eigenvalues, and small entries. Further, the eigenvalues of $A,C$ are quite similar -- one is near 5, one is near 0. – vadim123 Jul 08 '13 at 07:25
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    In short, not only is dropping $B$ a bad idea, even rounding the entries of your matrices might lead to catastrophic failure. – vadim123 Jul 08 '13 at 07:26
  • Thanks again. I now understand that if $C$ will get ill-conditioned by this approx. I will get severe problems. But in my case $C$ will be a positive semidefinite matrix (since $A$ is PSD). So, $C$ may turn by this approximation from a maybe indefinite matrix to a PSD matrix. This is also a reason for using this approximation. Can I now make a valid statement when I can use this approximation? – bonanza Jul 08 '13 at 08:26
  • @bonanza, please ask another question where you give all the specifics. – vadim123 Jul 08 '13 at 14:43