Mathematicians prove Pólya's conjecture for the eigenvalues of a disk
phys.org
phys.org
edit: Ah, I misunderstood the problem. The eigenfunctions are exactly solved; the problem of sorting and ordering their eigenvalues is apparently not! From their page 4,
- "Although all the eigenvalues of the Dirichlet and Neumann Laplacians on the unit disk are explicitly known in terms of zeros of the Bessel functions or their derivatives, see §2 below, in each case the spectrum is given by a two-parametric family, and rearranging it into a single monotone sequence appears to be an unfeasible task."
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.
It's rather remarkable, unexpected, and perhaps shocking, when you first hear it. There does not seem to be enough information to make it true. But it is.
I used it in a real project to cancel noise.
https://www.amazon.com/Matrix-Theory-Dover-Books-Mathematics...
https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010...
Given Ax=λx, take i for which |xᵢ| is largest. Look at the i'th equation: sum aᵢⱼxⱼ = λxᵢ, move the aᵢᵢxᵢ term to the rhs, take absolute values, divide by |xᵢ|, apply triangle inequality, and you have |aᵢᵢ - λ| ≤ sum |aᵢⱼ| over j≠i. So for every eigenvalue you can find such a disc.
That's by column, for row use Aᵀ.
If you had played around a bit with Laplacian matrices, like tri-diagonal matrices with stencil [-1, 2, -1], and found that its eigenvalues are within 2 ± 2, and if you also realized that A + τI has the same eigenvalues shifted by τ, then it's a small step to consider that the magnitude of the off-diagonal may have something to do with the spread of eigenvalues.
It's likely that Gerschgorin stumbled upon it like this.
Just trying to understand these terms. So is a 5x5 tridiagonal matrix with your stencil look like this?
2 -1 0 0 0
-1 2 -1 0 0
0 -1 2 -1 0
0 0 -1 2 -1
0 0 0 -1 2See https://en.wikipedia.org/wiki/Compact_stencil#Three_Point_St...
Math 101, simple is better.
A common noise cancellation technique is to throw away small eigenvalues, as in PCA. This result relates eigenvalues to the structure of the matrix, so might be helpful for reducing ev's without bothering with diagonalization?
[Edit] This would presumably involve just zeroing out the rows with small diagonal elements and small-ish off-diagonal norm... Center the eigenvalue estimate disk at zero, and then zero out the rest of the row to make the estimate exact.
> 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?
r_i = \sum_{i \ne j} |A_{i j}|The answer to the above question is Gerschgorin disks and it's closely related cousin Brauer's Oval of Cassini.
For matrices with real eigenvalues it's moreso along the real number line, only for cases where the eigenvalues are imaginary do we imagine disks.
That line got me as well.
Not a proof or anything, of course the proof is on Wikipedia and nice and elegant. Just a thought on the gut feeling.
I agree that it is a very nice result.
The diagonal elements of matrices have a lot of rather "magical" properties if you think about it. Their sum is also the sum of the eigenvalues of the matrix. And if you have a matrix A that is singular, you can choose any value x that is not an eigenvalue, and then A - xI is invertible but still mostly behaves like A.
"The celebrated Pólya’s conjecture (1954) in spectral geometry states that the eigenvalue counting functions of the Dirichlet and Neumann Laplacian on a bounded Euclidean domain can be estimated from above and below, respectively, by the leading term of Weyl’s asymptotics. Pólya’s conjecture is known to be true for domains which tile Euclidean space, and, in addition, for some special domains in higher dimensions. In this paper, we prove Pólya’s conjecture for the disk, making it the first non-tiling planar domain for which the conjecture is verified. We also confirm Pólya’s conjecture for arbitrary planar sectors, and, in the Dirichlet case, for balls of any dimension. Along the way, we develop the known links between the spectral problems in the disk and certain lattice counting problems. A key novel ingredient is the observation, made in recent work of the last named author, that the corresponding eigenvalue and lattice counting functions are related not only asymptotically, but in fact satisfy certain uniform bounds."
> Is it possible to deduce the shape of a drum from the sounds it makes?
That's a known problem with a (very nice) negative answer https://www.ams.org/publicoutreach/feature-column/fcarc-1997...
IIUC this article is about the problem in the other direction, i.e. from the shape (a disk!) to the frecuencies of the sound (eigenvalues).It's not about an exact calculation, but about an aproximation of them.
> The conjecture bears on the estimation of the frequencies of a round drum or, in mathematical terms, the eigenvalues of a disk.
From the research paper:
> The celebrated Pólya’s conjecture (1954) in spectral geometry states that the eigenvalue counting functions of the Dirichlet and Neumann Laplacian on a bounded Euclidean domain can be estimated from above and below, respectively, by the leading term of Weyl’s asymptotics.
<guessing> The Weyl's asymtotics is probably a good estimation of the very high frecuencies/eigenvalues, and the conjeture is probabbly that you can use the estimation as upper or lower bounds instead of just an aproximation.<guessing> [Sorry, not my area and I have not enough time to read the paper.]
That's a very different beast to the game mathematicians play, which demands rigorous proof (or at least a fairly close social version of it, it's turtles all the way down).