Gershgorin
Visualize the Gershgorin discs that bound the spectrum of a square matrix (see the Gershgorin disc theorem).
Installation
Hit ] in the Julia REPL to open the package manager, then use
pkg> add GershgorinQuickstart Guide
We can visualize the Gershgorin discs for a random complex matrix and its transpose. Note that a matrix and its transpose have the same eigenvalues, so these eigenvalues will lie in the intersection of the Gershgorin regions of these two matrices.
using LinearAlgebra
using Plots, LaTeXStrings
using Gershgorin
# Make a random (5,5) real matrix
M = randn(5, 5)
# Plot Gershgorin's discs
gershgorin(M; c=:blue, label=L"$M$")
λ = eigvals(M)
p1 = plot!(λ, seriestype=[:scatter], c=:black, label=L"$\lambda(M)$")
# Now do the same for the transpose
gershgorin(transpose(M); c=:red, label=L"$M^T$")
λ = eigvals(transpose(M))
p2 = plot!(λ, seriestype=[:scatter], c=:black, label=L"$\lambda(M^T)$")
# Plot the intersection between the two sets of regions
gershgorin(M; c=:blue)
gershgorin!(transpose(M); c=:red)
overlap(M, transpose(M), c=:black, alpha=1)
p3 = plot!(λ, seriestype=[:scatter], c=:black, label=L"$\lambda(M)$")
plot(p1, p2, p3, link=:all, dpi=300, layout=(1,3), size=(750,350))
Additionally, if you just want to get the Gershgorin discs for a matrix, you can use the get_discs function.
discs = get_discs(M)
plot(discs, c=:blue, alpha=0.2, lw=0)
plot!(eigvals(M), seriestype=[:scatter], c=:black, label=L"$\lambda(M)$", aspect_ratio=1) |> display