Compute the eigenvalues and eigenvectors of a matrix?

Questions
,
1

Hello all,

I have been having a lot of trouble computing the eigenvalues and eigenvectors of a matrix for a problem that I am working on in computational fluid dynamics. I am new to Mathematica so I am not very familiar with the syntax and I can not find out what is wrong with my code. I have uploaded my script so that you can see how I am trying to compute the eigenvalues and eigenvectors (at the very end of the script). Any help is appreciated, thanks.

Jacobians Eigenvalues and Eigenvectors for 3d Navier Stokes.nb (145 KB)

2

Your A is not a matrix.

Try

In[30]:= Dimensions[A]

Out[30]= {6, 1, 6, 1}

To compute Eigenvalues you have to have a (square) matrix, that is a list of (one-dimensional) lists.

Look at

In[31]:= m = {{a, b}, {c, d}};
Dimensions[m]
Eigenvalues[m]

Out[32]= {2, 2}

Out[33]= {1/2 (a + d - Sqrt[a^2 + 4 b c - 2 a d + d^2]), 
 1/2 (a + d + Sqrt[a^2 + 4 b c - 2 a d + d^2])}

Why do you write at the beginning

F= { {q2}, ...

putting your variables (?) between { }? That makes q2 to a List.

In[34]:= FullForm[{q2}]

Out[34]//FullForm= \!\(\*
TagBox[
StyleBox[
RowBox[{"List", "[", "q2", "]"}],
ShowSpecialCharacters->False,
ShowStringCharacters->True,
NumberMarks->True],
FullForm]\)
3

I made a couple of changes in your notebook and arrive at a list of Eigenvalues. Is it that what you meant?

Jacobians Eigenvalues and Eigenvectors for 3d Navier Stokes.nb (125 KB)

4

Yes, this is exactly what I meant. Sorry, I am really new to mathematica and I was just getting confused with the syntax. I really appreciate your help. Is there any way you could tell me how to compute the eigenvectors as well? What I am trying to do is compute the eigenvectors (hopefully one eigenvector for each eigenvalue) and put them in a matrix where each column is an eigenvector. Does this make sense?

5

Try

Eigensystem[ matrix ]

That gives you a list. 1st entry is the list of eigenvalues, 2nd entry is a list of Eigenvectors.

And I think you want

Transpose[Eigenvectors[ matrix ] ]

6

Thank you so much.

Is this correct? I’m having trouble getting output.

Jacobians Eigenvalues and Eigenvectors for 3d Navier Stokes.nb (129 KB)

7

Well, look at your expressions in matrix A. And A is 6 x 6 - Matrix. It seems to be utmost tedious if not impossible to obtain closed expressions for the eigenvectors. Try to do it numerically.

8

After quite a time of computation mma comes back with explicit expressions for the Eigenvectors. Note that your systems is four-fold degenerate.

The 1st four Eigenvectors are “simple”, the next two (belonging to the Eigenvlaues with square-root) by far not.

See notebook (at the end).

Jacobians Eigenvalues and Eigenvectors for 3d Navier Stokes _OP_1_CP_.nb (181 KB)

9

I really appreciate your help, thank you so much. I now know that I will more than likely have to compute these numerically in my Fortran code. Thanks once again

10

No need to recur to FORTRAN: Just feed numbers in your mma-expression.
E.g.

A1= A/.{gamma->.5, x1x->7, … and so on

You could as well try to find abbreviations for the matrix-elements of A, that could make things easier to handle.

11

Abbreviations is a good idea, I may try that. I need to eventually put it into fortran because I am working with a computational fluid dynamics solver which is written in fortran. I am trying to let mathematica do the derivation of a physical problem for me so that I can add it to a pre-existing code. Ultimately, the eigenvector calculation will need to be done thousands of times as I step through the CFD solution.

12

At least you have more or less simple expressions for 4 out of 6 Eigenvectors.

In case you don’t know about it try

FortranForm[ eigenvector ]

I played a bit around with abbreviations in matrix A, but had no real success in making things easier to handle.

I think mma will calculate eignevectors in quite short a time if you feed numbers into matrix A, and as far as I know it is possible to connect mma to Fortran.

Kind regards HD