Timeline for Formula or estimates for $\frac{\operatorname{Tr}(AB)}{\operatorname{Tr}(A)}$
Current License: CC BY-SA 3.0
15 events
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| Jan 18, 2017 at 19:34 | history | edited | Ian | CC BY-SA 3.0 |
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| Jan 18, 2017 at 19:15 | history | edited | Ian | CC BY-SA 3.0 |
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| Jan 18, 2017 at 19:06 | history | edited | Ian | CC BY-SA 3.0 |
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| Jan 18, 2017 at 13:22 | comment | added | Steve Huntsman | This is the fundamental computational problem of thermal quantum field theory: en.wikipedia.org/wiki/Thermal_quantum_field_theory | |
| Jan 17, 2017 at 15:48 | comment | added | Ian | @NikWeaver That's an interesting way to think about it. (I've also edited in the detail that my matrices have nonnegative real entries.) | |
| Jan 17, 2017 at 15:48 | history | edited | Ian | CC BY-SA 3.0 |
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| Jan 17, 2017 at 15:26 | comment | added | Nik Weaver | It's unclear what you're looking for, but if you give the symmetric $n\times n$ matrices Hilbert-Schmidt norm they form an inner product space and the ratio you are interested in is $\frac{\langle A,B\rangle}{\langle A,I\rangle}$. Would that help? (I'm assuming you are talking about real matrices.) | |
| Jan 17, 2017 at 14:46 | history | edited | Ian | CC BY-SA 3.0 |
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| Jan 17, 2017 at 14:42 | comment | added | Ian | @Dirk This is a good point. In fact the latter form is how my question came about in the first place; I expected that this permutation would be easier to manage because of the symmetry of $B$. | |
| Jan 17, 2017 at 14:42 | comment | added | Ian | @CarloBeenakker The entrywise formula, i.e. $\frac{\sum_{i=1}^n \sum_{j=1}^n a_{ij} b_{ij}}{\sum_{i=1}^n a_{ii}}$, is not really good enough, because in my context $n$ is going to infinity. So I'm looking for something I can use to estimate this ratio in this situation. | |
| Jan 17, 2017 at 14:28 | comment | added | Carlo Beenakker | the trivial computational complexity is $n^2$, is that efficient enough? | |
| Jan 17, 2017 at 14:08 | comment | added | Dirk | Note that by $\operatorname{tr}(AB) = \operatorname{tr}(AD^TD) = \operatorname{tr}(DAD^T)$ the question is equivalent to the question on the relation between the trace of a matrix and a congruent one - unfortunately I don't know the either to that question either. | |
| Jan 17, 2017 at 13:48 | history | edited | Ian | CC BY-SA 3.0 |
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| Jan 17, 2017 at 13:45 | review | First posts | |||
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| Jan 17, 2017 at 13:42 | history | asked | Ian | CC BY-SA 3.0 |