4 ms·
I think that's just how textbooks present it. A fact is stated but you're lacking intuition. If you had played around a bit with Laplacian matrices, like tri-d
by stabbles 3y ago
I think that's just how textbooks present it. A fact is stated but you're lacking intuition.
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.
- xelxebar 3y ago> tri-diagonal matrices with stencil [-1, 2, -1], 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 2
- stabbles 3y agoYeah, exactly. Your matrix shows up when you do a 3-point discretization of -u''(x) = λu(x) with u(0) = u(1) = 0, at 5 equidistant (interior) grid points. That's the discrete, 1D version of the problem they're looking at in the paper. See https://en.wikipedia.org/wiki/Compact_stencil#Three_Point_Stencil_Example https://en.wikipedia.org/wiki/Compact_stencil#Three_Point_St...