Timeline for square matrix depending on complex value: spectral radius continous?
Current License: CC BY-SA 4.0
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| Sep 6, 2023 at 17:43 | comment | added | Iosif Pinelis | @JochenGlueck : This is a good point! | |
| Sep 6, 2023 at 17:02 | comment | added | Jochen Glueck | @MikhailKatz: Using just Gelfand's formula can't do the job since Gelfand's formula also holds in infinite dimensions while contuity of the spectral radius does not. | |
| Sep 6, 2023 at 15:14 | comment | added | Iosif Pinelis | @MikhailKatz : (i) I think the argument is already very simple. (ii) To deal with the maximum of the moduli of all the eigenvalues, we need to deal with all the eigenvalues somehow. Of course, enumerating them is only one way to do that. Alternatively, one can e.g. describe the continuity of the set of eigenvalues the way it is done in Section 5.1 of Chapter II of Kato's book, linked in my answer. (iii) Like you, I don't see a simple (or any other) way to use Gelfand's formula here. | |
| Sep 6, 2023 at 14:37 | comment | added | Mikhail Katz | By Gelfand's formula, the spectral radius can be computed as the limit of operator norms of the powers of the matrix, but I don't see immediately how to derive continuity. There should be a simpler argument than enumerating all the eigenvalues. | |
| Sep 6, 2023 at 13:40 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Sep 6, 2023 at 13:03 | history | edited | Brendan McKay | CC BY-SA 4.0 |
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| Sep 6, 2023 at 12:58 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Sep 6, 2023 at 12:49 | history | answered | Iosif Pinelis | CC BY-SA 4.0 |