It’s not so typically in numerical linear algebra that the plots we produce are visually engaging. This plot got here up in some MATLAB experiments. Are you able to guess what it’s?
I plotted the Gershgorin discs of a stochastic matrix: a matrix with nonnegative components and row sums all equal to . The Gershgorin discs for an
matrix
are the
discs within the advanced aircraft outlined by
Gershgorin’s theorem says that the eigenvalues of lie within the union of the discs.
Why do the discs kind this attention-grabbing sample? For a stochastic matrix the th Gershgorin disc is
This disc goes via and the nearer
is to
the smaller the radius of the disc, so the discs are nested, with the disc similar to
containing all of the others.
The matrix used for the plot is A = anymatrix('core/symmstoch',64)
from the Anymatrix assortment. It has diagonal components roughly uniformly distributed on , so the facilities of the discs are roughly equally spaced and shrink because the facilities transfer to the precise.
The picture above is for the matrix of dimension . The black dots are the eigenvalues. Right here is the plot for
. The perform used to supply these plots is
gersh
from the Matrix Computation Toolbox.
Listed here are two different matrices whose Gershgorin discs make a graphically attention-grabbing plot.
If you realize of another attention-grabbing examples please put them within the feedback beneath.