# Estimating the success of an approximation of a known matrix

I am trying to approximate a known $$N\times N$$ matrix $$A$$ with an 'estimation' matrix $$A'$$. The question is, how is it possible to quantify the error in this approximation - the difference between $$A$$ and $$A'$$.

If $$A$$ and $$A'$$ were single numbers, this problem would be trivial. However, when the data is in matrix form, the meaning of error becomes more ambiguous.

I am aware of RMS and matrix norms, but there is a problem with using these to quantify the estimation error as they are dependent on the dynamic range of the data - if the matrix values are in the range $$0-1000$$, an RMS value of $$0.5$$ does not tell anything useful; there could be significant differences between the matrix elements $$a_{ij}$$ and $$a_{ij}'$$, no matter what the RMS is. I think the matrix norm encounters the same problem exactly.

• This can be answered only by supplying additional information: what is the objective of this approximation? What is the cost of making an error of approximation and how would you quantify that error?
– whuber
Commented Aug 1 at 15:23

$$\frac{a_{ij}-a'_{ij}}{a_{ij}}$$
and to average these relative errors for all values. As mentioned in the comment, depending on the exact nature of your problem, all elements in the matrix may not need to be equally well estimated. For instance, if $$A$$ is a transition matrix in a dynamical state-space system, then it is more important to correctly estimate the high values than the low ones, and the classical Frobenius norm could be used, see for instance section Evaluation metrics for learning of model parameters in numerical simulations of the following paper: