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I have two matrices ${\bf H_1}$ and ${\bf H_2}$, Id like to create a metric that describes how close there singular values are assume ${\bf H_1}$ has eigen values $\lambda_{11}$ and $\lambda_{12}$ in decreasing order and ${\bf H_2}$ has eigen values $\lambda_{21}$ and $\lambda_{22}$ in decreasing order,, can i say a good metric is

$$\frac{|\lambda_{11}- \lambda_{12}| + |\lambda_{21}- \lambda_{22}| }{\lambda_{11}+\lambda_{12}}$$

And if it makes sense does it mean the closer this metric is to zero the more the matrices' eigenvalues match? Also is this metric between 0 and 1?

THANKS

Tyrone
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    What if $\lambda_{11} = - \lambda_{12}$? – qualcuno Aug 30 '18 at 03:30
  • Oops I didnt think of that ;( – Tyrone Aug 30 '18 at 03:31
  • Can you please help me create a reasonable metric – Tyrone Aug 30 '18 at 03:43
  • I don't know much about functional analysis, and I don't know what is motivating your question, but if by metric you refer to the metric spaces' concept of distance, you will need an operation that verifies $d(A,B) = 0$ if and only if $A = B$. Two matrices can have the same eigenvalues, and still be very different. – qualcuno Aug 30 '18 at 04:05
  • Thanks. I knw that the matrices can be very different and have same eigen values..but actually my application cares about the eigen values only.. I'm working on wireless communication problem and eigen values represent kind of power of channel – Tyrone Aug 30 '18 at 04:08
  • I see. Again, I'm far from knowledgeable in the subject, but maybe you are interested on a pseudodistance, that is, a distance with the exception that $d(A,B) = 0$ may not imply $A = B$ (the converse does however hold). – qualcuno Aug 30 '18 at 04:18

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