matrix - Mixing up eigenvalues with positive and negative imaginary parts

f[k_] = {{-0.001 - 2 I k, 1, -0.001, -0.501}, {0.001, -0.5 - I k, 0.001, 0.001},
     {-0.001, -0.501, -0.001 + 2 I k, 1.}, {0.001, 0.001, 0.001, -0.5 + I k}};

y[k_] = Im[Eigenvalues[f[k]][[3]]];
Plot[y[k], {k, -1, 1}]

3rd and 4th eigenvalues are complex conjugates, as we can see from the graph the program mixed them, there are lots of jump discontinuities from the one with negative imaginary part to the one with positive. I suspect that the program sometimes takes as the 3rd eigenvalue (or root) the one with positive "Im" and sometime the one with negative "Im".

Please help me to resolve this problem by a general approach (of course for this case one can take the Max of two as k>0 and Min otherwise to obtain the imaginary part of the desired eigenvalue).

Answer

I'm not sure exactly which eigenvalues you wish to display, but you can sort them in order to choose. For instance, to plot the largest eigenvalue, you can

f[k_] := {{-0.001 - 2 I k, 1, -0.001, -0.501}, {0.001, -0.5 - I k, 
    0.001, 0.001}, {-0.001, -0.501, -0.001 + 2 I k, 1.}, {0.001, 
    0.001, 0.001, -0.5 + I k}};

y[k_] := Sort[Im[Eigenvalues[f[k]]]][[3]];
Plot[y[k], {k, -1, 1}]

which gives

enter image description here

Sort also takes an optional second argument which can be used to pick out which of the eigenvalues you want plotted.

Comments

plotting - Filling between two spheres in SphericalPlot3D

Manipulate[ SphericalPlot3D[{1, 2 - n}, {θ, 0, Pi}, {ϕ, 0, 1.5 Pi}, Mesh -> None, PlotPoints -> 15, PlotRange -> {-2.2, 2.2}], {n, 0, 1}] I cant' seem to be able to make a filling between two spheres. I've already tried the obvious Filling -> {1 -> {2}} but Mathematica doesn't seem to like that option. Is there any easy way around this or ... Answer There is no built-in filling in SphericalPlot3D . One option is to use ParametricPlot3D to draw the surfaces between the two shells: Manipulate[ Show[SphericalPlot3D[{1, 2 - n}, {θ, 0, Pi}, {ϕ, 0, 1.5 Pi}, PlotPoints -> 15, PlotRange -> {-2.2, 2.2}], ParametricPlot3D[{ r {Sin[t] Cos[1.5 Pi], Sin[t] Sin[1.5 Pi], Cos[t]}, r {Sin[t] Cos[0 Pi], Sin[t] Sin[0 Pi], Cos[t]}}, {r, 1, 2 - n}, {t, 0, Pi}, PlotStyle -> Yellow, Mesh -> {2, 15}]], {n, 0, 1}]

plotting - Plot 4D data with color as 4th dimension

I have a list of 4D data (x position, y position, amplitude, wavelength). I want to plot x, y, and amplitude on a 3D plot and have the color of the points correspond to the wavelength. I have seen many examples using functions to define color but my wavelength cannot be expressed by an analytic function. Is there a simple way to do this? Answer Here a another possible way to visualize 4D data: data = Flatten[Table[{x, y, x^2 + y^2, Sin[x - y]}, {x, -Pi, Pi,Pi/10}, {y,-Pi,Pi, Pi/10}], 1]; You can use the function Point along with VertexColors . Now the points are places using the first three elements and the color is determined by the fourth. In this case I used Hue, but you can use whatever you prefer. Graphics3D[ Point[data[[All, 1 ;; 3]], VertexColors -> Hue /@ data[[All, 4]]], Axes -> True, BoxRatios -> {1, 1, 1/GoldenRatio}]

plotting - Mathematica: 3D plot based on combined 2D graphs

I have several sigmoidal fits to 3 different datasets, with mean fit predictions plus the 95% confidence limits (not symmetrical around the mean) and the actual data. I would now like to show these different 2D plots projected in 3D as in but then using proper perspective. In the link here they give some solutions to combine the plots using isometric perspective, but I would like to use proper 3 point perspective. Any thoughts? Also any way to show the mean points per time point for each series plus or minus the standard error on the mean would be cool too, either using points+vertical bars, or using spheres plus tubes. Below are some test data and the fit function I am using. Note that I am working on a logit(proportion) scale and that the final vertical scale is Log10(percentage). (* some test data *) data = Table[Null, {i, 4}]; data[[1]] = {{1, -5.8}, {2, -5.4}, {3, -0.8}, {4, -0.2}, {5, 4.6}, {1, -6.4}, {2, -5.6}, {3, -0.7}, {4, 0.04}, {5, 1.0}, {1, -6.8}, {2, -4.7}, {3, -1.

Blog

Search This Blog