Timeline for Sufficient conditions for all eigenvalues simple in stochastic matrix
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| S Jul 30, 2018 at 1:45 | history | suggested | Rodrigo de Azevedo | CC BY-SA 4.0 |
Minor improvements.
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| Jul 29, 2018 at 20:58 | review | Suggested edits | |||
| S Jul 30, 2018 at 1:45 | |||||
| Jul 29, 2018 at 18:41 | comment | added | Mateusz Kwaśnicki | @Randomguy: Just as for matrices: one requires that the arguments are arranged in an increasing order; in your example, $x_1 < x_2$. I do not think total positivity has been applied to study simplicity of eigenvalues in continuous case. Actually, when I wrote my first comment, I figured out that one should perhaps try this approach. Unfortunately, I believe that most interesting kernels are not totally positive. | |
| Jul 29, 2018 at 12:48 | comment | added | Federico Poloni | Also: why do you need to prove that all eigenvalues are simple? Probably it's worth asking yourself if the result that you need holds also for matrices with non-simple eigenvalues... | |
| Jul 29, 2018 at 10:05 | comment | added | Jochen Glueck | If the matrix is irreducible, then not only the eigenvalue $1$, but all eigenvalus of modulus $1$ are simple. For the eigenvalues of smaller modulus I second @MateuszKwaśnicki's comment that the question is likely to be very difficult to answer. | |
| Jul 29, 2018 at 9:21 | comment | added | Mateusz Kwaśnicki | Similar problems in continuous variable (that is, for stochastic kernels rather than stochastic matrices) are considered to be difficult, so I doubt there are general methods to show simplicity of eigenvalues. The only thing that comes to my mind is a result for strictly totally positive matrices; see Section 6 here. | |
| Jul 29, 2018 at 9:05 | review | First posts | |||
| Jul 29, 2018 at 9:13 | |||||
| Jul 29, 2018 at 9:02 | history | asked | Randomguy | CC BY-SA 4.0 |