Mathematicians prove Pólya's conjecture for the eigenvalues of a disk

Post by frutiger » Sat, Mar 02, 2024, 4:00 PM UTC

mikhailfranco wrote: post #9

The title words remind of an unrelated fact, Gershgorin Disks: The eigenvalues of any N x N matrix, A, are contained in the union of N discs in the complex plane. The center of the i_th disc is the i_th diagonal element of A. The radius of the i_th disc is the absolute values of the off-diagonal elements in the i_th row. https://blogs.sas.com/content/iml/2019/05/22/gershgorin-disc... It's rather remarkable, unexpecte…

Post by reactordev » Sat, Mar 02, 2024, 4:05 PM UTC

frutiger wrote: post #11

mikhailfranco wrote: post #9

The title words remind of an unrelated fact, Gershgorin Disks: The eigenvalues of any N x N matrix, A, are contained in the union of N discs in the complex plane. The center of the i_th disc is the i_th diagonal element of A. The radius of the i_th disc is the absolute values of the off-diagonal elements in the i_th row. https://blogs.sas.com/content/iml/2019/05/22/gershgorin-disc... It's rather remarkable, unexpecte…

Sounds very cool. One thing I didn’t understand though: > The radius of the i_th disc is the absolute values How can a radius of a single disc (i.e. a single value) correspond to multiple values?

Post by abetusk » Sat, Mar 02, 2024, 4:18 PM UTC

ximeng wrote: post #14

frutiger wrote: post #11

Earlier quoted context omitted.

Sounds very cool. One thing I didn’t understand though: > The radius of the i_th disc is the absolute values How can a radius of a single disc (i.e. a single value) correspond to multiple values?

If you look at the link there’s a formula showing it’s the sum.

    r_i = \sum_{i \ne j} |A_{i j}|

Post by Sharlin » Sat, Mar 02, 2024, 4:23 PM UTC

reactordev wrote: post #13

frutiger wrote: post #11

Earlier quoted context omitted.

Sounds very cool. One thing I didn’t understand though: > The radius of the i_th disc is the absolute values How can a radius of a single disc (i.e. a single value) correspond to multiple values?

I’m assuming, but could be wrong as my matrix math ended with GameDev, that the radius of the disk = the absolute values of the other things. +\- depending on which side of the disk they fall? That line got me as well.

Post by eigenket » Sat, Mar 02, 2024, 4:40 PM UTC

esafak wrote: post #10

I haven't read the paper, but does it apply to arbitrarily shaped drums? And if the basis can be arbitrarily large, is this an exciting result? It just says that the spectral domain is isomorphic?