Is Mathematica the best option to compute the eigenvalues?

[JD_PM]

TL;DR Summary

I am wondering what's the best option to compute the eigenvalues for such a determinant

I am wondering what's the best option to compute the eigenvalues for such a determinant$$\begin{vmatrix}
\sin \Big( n \frac{\omega}{v_1} \theta \Big) & \cos \Big( n \frac{\omega}{v_1} \theta \Big) & 0 & 0 \\
0 & 0 & \sin \Big( n \frac{\omega}{v_2} (2 \pi - \theta) \Big) & \cos \Big( n \frac{\omega}{v_2} (2 \pi - \theta) \Big) \\
\sin \Big( n \frac{\omega}{v_1} \pi \Big) & \cos \Big( n \frac{\omega}{v_1} \pi \Big) & -\sin \Big( n \frac{\omega}{v_2} \pi \Big) & -\cos \Big( n \frac{\omega}{v_2} \pi \Big) \\
\frac{n}{v_1} \cos \Big( n \frac{\omega}{v_1} \pi \Big) & -\frac{n}{v_1} \sin \Big( n \frac{\omega}{v_1} \pi \Big) & \frac{n}{v_2} \cos \Big( n \frac{\omega}{v_2} \pi \Big) & -n\frac{B_2}{v_2} \sin \Big( n \frac{\omega}{v_2} \pi \Big) \\
\end{vmatrix} = 0$$Where $L = R \theta$ and L and R are known.

I was looking at mathematica:

https://reference.wolfram.com/language/tutorial/EigenvaluesAndEigenvectors.html

But the examples they show are much more simpler than the one I am dealing with,

Should I use Matlab instead?

Any suggestion is appreciated.

If you are interested from where is this coming from check this out: https://math.stackexchange.com/ques...osed-loop?noredirect=1#comment7161084_3482613

 

[Bill Simpson]

Let's suppose, just as an experiment, you wanted the eigenvalues of a matrix

Eigenvalues[{{Sin[n w/v1 t],Cos[n w/v1 t],0,0},
{0,0,Sin[n w/v2(2Pi-t)],Cos[n w/v2(2Pi-t)]},
{Sin[n w/v1 Pi],Cos[n w/v1 Pi],-Sin[n w/v2 Pi],-Cos[n w/v2 Pi]},
{n/v1 Cos[n w/v1 Pi],-n/v1 Sin[n w/v1 Pi],n/v2 Cos[n w/v2 Pi],-n b2/v2 Sin[n w/v2 Pi]}}]

Then Mathematica would respond with a vector of four expressions, the first expression is

Root[-(n*v1^4*v2^3*Sin[(n*Pi*w)/v1 - (n*t*w)/v1 - (n*Pi*w)/v2 - (n*(2*Pi - t)*w)/v2]) + b2*n*v1^4*v2^3*
  Sin[(n*Pi*w)/v1 - (n*t*w)/v1 - (n*Pi*w)/v2 - (n*(2*Pi - t)*w)/v2] - n*v1^4*v2^3*Sin[(n*Pi*w)/v1 -
  (n*t*w)/v1 + (n*Pi*w)/v2 - (n*(2*Pi - t)*w)/v2] - b2*n*v1^4*v2^3*Sin[(n*Pi*w)/v1 - (n*t*w)/v1 +
  (n*Pi*w)/v2 - (n*(2*Pi - t)*w)/v2] - 2*n*v1^3*v2^4*Sin[(n*Pi*w)/v1 - (n*t*w)/v1 + (n*Pi*w)/v2 -
  (n*(2*Pi - t)*w)/v2] - n*v1^4*v2^3*Sin[(n*Pi*w)/v1 - (n*t*w)/v1 - (n*Pi*w)/v2 + (n*(2*Pi - t)*w)/v2] -
  b2*n*v1^4*v2^3*Sin[(n*Pi*w)/v1 - (n*t*w)/v1 - (n*Pi*w)/v2 + (n*(2*Pi - t)*w)/v2] +
  2*n*v1^3*v2^4*Sin[(n*Pi*w)/v1 - (n*t*w)/v1 - (n*Pi*w)/v2 + (n*(2*Pi - t)*w)/v2] - n*v1^4*v2^3*
  Sin[(n*Pi*w)/v1 - (n*t*w)/v1 + (n*Pi*w)/v2 + (n*(2*Pi - t)*w)/v2] + b2*n*v1^4*v2^3*Sin[(n*Pi*w)/v1 -
  (n*t*w)/v1 + (n*Pi*w)/v2 + (n*(2*Pi - t)*w)/v2] + (-2*n*v1^2*v2^3*Cos[(n*Pi*w)/v1 - (n*t*w)/v1 -
  (n*(2*Pi - t)*w)/v2] - 2*v1^3*v2^3*Cos[(n*Pi*w)/v1 - (n*t*w)/v1 - (n*(2*Pi - t)*w)/v2] -
  n*v1^3*v2^2*Cos[(n*Pi*w)/v1 - (n*Pi*w)/v2 - (n*(2*Pi - t)*w)/v2] + b2*n*v1^3*v2^2*Cos[(n*Pi*w)/v1 -
  (n*Pi*w)/v2 - (n*(2*Pi - t)*w)/v2] - n*v1^3*v2^2*Cos[(n*Pi*w)/v1 + (n*Pi*w)/v2 - (n*(2*Pi - t)*w)/v2] -
  b2*n*v1^3*v2^2*Cos[(n*Pi*w)/v1 + (n*Pi*w)/v2 - (n*(2*Pi - t)*w)/v2] - 2*n*v1^2*v2^3*Cos[(n*Pi*w)/v1 +
  (n*Pi*w)/v2 - (n*(2*Pi - t)*w)/v2] - 2*n*v1^2*v2^3*Cos[(n*Pi*w)/v1 - (n*t*w)/v1 + (n*(2*Pi - t)*w)/v2] +
  2*v1^3*v2^3*Cos[(n*Pi*w)/v1 - (n*t*w)/v1 + (n*(2*Pi - t)*w)/v2] - n*v1^3*v2^2*Cos[(n*Pi*w)/v1 -
  (n*Pi*w)/v2 + (n*(2*Pi - t)*w)/v2] - b2*n*v1^3*v2^2*Cos[(n*Pi*w)/v1 - (n*Pi*w)/v2 + (n*(2*Pi - t)*w)/v2] +
  2*n*v1^2*v2^3*Cos[(n*Pi*w)/v1 - (n*Pi*w)/v2 + (n*(2*Pi - t)*w)/v2] - n*v1^3*v2^2*Cos[(n*Pi*w)/v1 +
  (n*Pi*w)/v2 + (n*(2*Pi - t)*w)/v2] + b2*n*v1^3*v2^2*Cos[(n*Pi*w)/v1 + (n*Pi*w)/v2 + (n*(2*Pi - t)*w)/v2] -
  2*n*v1^3*v2^2*Sin[(n*t*w)/v1] - 2*b2*n*v1^3*v2^2*Sin[(n*t*w)/v1] - n*v1^3*v2^2*Sin[(n*t*w)/v1 -
  (2*n*Pi*w)/v2] + b2*n*v1^3*v2^2*Sin[(n*t*w)/v1 - (2*n*Pi*w)/v2] - n*v1^3*v2^2*Sin[(n*t*w)/v1 +
  (2*n*Pi*w)/v2] + b2*n*v1^3*v2^2*Sin[(n*t*w)/v1 + (2*n*Pi*w)/v2])*#1 + (2*n*v1^2*v2 + 2*b2*n*v1^2*v2 +
  2*n*v1^2*v2*Cos[(2*n*Pi*w)/v2] - 2*b2*n*v1^2*v2*Cos[(2*n*Pi*w)/v2] - 2*b2*n*v1^2*v2*Cos[(n*t*w)/v1 -
  (n*Pi*w)/v2] - 2*v1^2*v2^2*Cos[(n*t*w)/v1 - (n*Pi*w)/v2] + 2*b2*n*v1^2*v2*Cos[(n*t*w)/v1 + (n*Pi*w)/v2] +
  2*v1^2*v2^2*Cos[(n*t*w)/v1 + (n*Pi*w)/v2] + 2*n*v1*v2^2*Sin[(n*Pi*w)/v1 - (n*(2*Pi - t)*w)/v2] +
  2*v1^2*v2^2*Sin[(n*Pi*w)/v1 - (n*(2*Pi - t)*w)/v2] + 2*n*v1*v2^2*Sin[(n*Pi*w)/v1 + (n*(2*Pi - t)*w)/v2] -
  2*v1^2*v2^2*Sin[(n*Pi*w)/v1 + (n*(2*Pi - t)*w)/v2])*#1^2 + (-4*v1*v2*Sin[(n*t*w)/v1] +
  4*b2*n*v1*Sin[(n*Pi*w)/v2] + 4*v1*v2*Sin[(n*Pi*w)/v2])*#1^3 + 4*#1^4 & , 1]/(v1*v2),

The remaining three expressions are similar to the first.

You can look up Root in the help system to see what that is. Usually Root[expr] is more compact than the expanded analytic expression, if one is even available.

 

[JD_PM]

Let's suppose, just as an experiment, you wanted the eigenvalues of a matrix

Thanks for your reply.

I've been trying to compute the eigenvalues but there's something going wrong. I've copied/pasted the code you provided and I got a wrong output.

But you indeed got a good output.

What am I missing?

 

[JD_PM]

OK I confused WolframAlpha with WolframMathematica.

Now it is clear.