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I have a matrix that I obtained from theoretical computation and I have another matrix which I obtained by actual data collection. How do I compare the two matrices? How do I state that one matrix is significantly different from the other?

I have looked into:

  • Frobenius Norm but I don't know what should be my threshold for comparison.
  • Looking at max and min deviation of one matrix from the other. But again, what's a good threshold?

I thought that to say one matrix varies significantly from the other, I would need to quantify the variance of those terms and compare how the deviation compares to the variance (or SD) but sadly, I don't have data for that and neither can I theoretically estimate any variances.

I am kind of lost in what would be a good metric.

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This depends entirely on your application. Different metrics will be appropriate for different purposes. –  Chris Eagle Nov 29 '12 at 17:56
    
Mathematical Statistics. –  Inquest Nov 29 '12 at 18:01
    
How was the matrix constructed and what do its entries represent? –  Fly by Night Nov 29 '12 at 18:59
    
@FlybyNight. The theoretical matrix is constructed using regression and the obtained matrix is well, obtained. –  Inquest Nov 29 '12 at 19:54
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In that case, the metric will be constructed using metric theory and it's details will be found, somehow. –  Fly by Night Nov 29 '12 at 21:27
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