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I have a Matrix, which is the sum of many other matrices:

$\boldsymbol C = \sum\limits _{i=1}^{n} ( \boldsymbol A_i + \boldsymbol B_i ) = \boldsymbol{\tilde{A}} + \sum\limits_{i=1}^{n} \boldsymbol B_i$

Now, I want to investigate whether the approximation $\boldsymbol C\approx \boldsymbol{\tilde{A}}$ is good.

What could be suitable criteria for the analysis of this?

So far I thought about the norm of the eigenvalues of the matrices $\boldsymbol B_i$ and the norm of the matrices $\boldsymbol B_i$ it self. What do you think?

Thank you very much in advance!

EDIT: I think I'm unable to explain precisely what I mean. In the end it all comes to the question in which cases the sum: $ \boldsymbol C = \boldsymbol A + \boldsymbol B$ could be approximated by: $ \boldsymbol C \approx \boldsymbol A$ And what could be the criteria to say that this approximation is good and valid.

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  • $\begingroup$ Can you specify for what purpose you are computing $C$? This matters in order to define a "good approximation". $\endgroup$ – Eric Kightley Jul 6 '13 at 13:33
  • $\begingroup$ C is in fact the Hessian of an other function. And $\tilde{A}$ is $J^TJ$, where $J$ denotes the Jacobian of this function. Later I have to invert matrix $C$. $\endgroup$ – bonanza Jul 6 '13 at 13:44
  • $\begingroup$ It is still not clear which norm we should use, or why we would prefer the eigenvalues of $\mathbf{B}_i$ over such a norm. Also, I presume that the $\mathbf{B}_i$ are unknown (or you would just compute $\mathbf{C}$ exactly, and so how are we going to compute their norms or eigenvalues? $\endgroup$ – Eric Kightley Jul 6 '13 at 13:48
  • $\begingroup$ thanks again for your reply. Yes, the main question is under which circumstances you would say that the sum over $\pmb B_i$ is negligible for $\pmb C$, compared to $\tilde{A}$. Furthermore, I can analytically calculate the norm and eigenvalues of $B_i$. $\endgroup$ – bonanza Jul 6 '13 at 15:04
  • $\begingroup$ @bonanza: You have not answered the question: the sum is neglible in what respect exactly? $\endgroup$ – tomasz Jul 6 '13 at 15:27

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