# Calculating eigenvalues of one matrix

1. Nov 13, 2012

### quin

hi friends

i want to find eigenvalues of a 4*4 matrix but niether MATLAB nor MATHEMATICA cant solve it. Both of programs said that eiganvalues of matrix is too complicated and have infinite sentences.
now what can i do?is there anyway that simplify the steps for matlab or mathematica?

the matrix is here:

#### Attached Files:

• ###### eigenvalues.m
File size:
681 bytes
Views:
74
2. Nov 14, 2012

### Bill Simpson

Mathematica gives four eigenvalues in a few seconds for

a = {{0, 1 + E^(-I*x) + E^(-I*z) + E^(-I*(x + z)), 1 + E^(-I*x) + E^(-I*y) + E^(-I*(x + y)), 1 + E^(-I*y) + E^(-I*z) + E^(-I*(y + z))},
{1 + E^(I*x) + E^(I*z) + E^(I*(x + z)), 0, 1 + E^(-I*y) + E^(I*z) + E^(-I*(y - z)), 1 + E^(I*x) + E^(-I*y) + E^(I*(x - y))},
{1 + E^(I*x) + E^(I*y) + E^(I*(x + y)), 1 + E^(I*y) + E^(-I*z) + E^(I*(y - z)), 0, 1 + E^(I*x) + E^(-I*z) + E^(I*(x - z))},
{1 + E^(I*y) + E^(I*z) + E^(I*(y + z)), 1 + E^(-I*x) + E^(I*y) + E^(-I*(x - y)), 1 + E^(-I*x) + E^(I*z) +E^(-I*(x - z)), 0}};
Eigenvalues[a]

Each of those is a root of a quartic equation. Using ToRadicals on each of those will give you the explicit solution

3. Nov 14, 2012

### quin

thank you so much dear it worked

But I have another question too
I found 4 eigenvalues and they are 4 sentences in terms of x,y,z

now I wanna expand all of them (4 of them) for "small x and small y and small z"

can you give me the the suitable formula for mathematica for small argument expansion?

thank you

Last edited: Nov 14, 2012
4. Nov 14, 2012

### Bill Simpson

If you had an "ordinary" expression then

Limit[Limit[Limit[p, x -> 0], y -> 0], z -> 0]

would give you the limit as your three variables go to zero, but Mathematica ToRules and Limit doesn't seem happy that everything you have is in terms of of complex exponentials.

Eigenvalues[a] //. {x -> 10^-9, y -> 10^-9, z -> 10^-9}

returns

{-2 - E^(-I/1000000000) - E^(I/1000000000),
-2 - E^(-I/1000000000) - E^(I/1000000000),
-2 - E^(-I/1000000000) - E^(I/1000000000),
3*(2 + E^(-I/1000000000) + E^(I/1000000000))}

BUT that is only approaching zero from one special direction and that direction is going to result in a lot of cancellations in your eigenvalues.

Perhaps you need to think carefully about exactly what you are trying to accomplish. That might give you an idea of what direction you want to approach this from. No pun intended.

Last edited: Nov 14, 2012
5. Nov 15, 2012