Homework Help: Finding eigenvalues and eigenvectors

1. Jan 8, 2006

fargoth

is there any trick for finding the eigenvalues and vectors for this kind of matrix?
$$\left( \begin{array}{ccccc} 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & \sqrt{\frac{3}{2} & 0 & 0 \\ 0 & \sqrt{\frac{3}{2} & 0 & \sqrt{\frac{3}{2} & 0 \\ 0 & 0 & \sqrt{\frac{3}{2} & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 \\ \end{array} \right)$$

i mean, i can tell the eigenvalues are 2,1,0,-1,-2... and i can tell the eigenvectors would have a=e and b=d... but thats because i know what this matrix is... but if i'll see some matrix with different values then this roaming around.... i dont know what i'll do, i dont think trying to solve the standard polynom of it is a good idea... and after knowing the eigenvalues one has to solve the set of equations to find the eigenvectors -yuck!-

Last edited: Jan 8, 2006
2. Jan 8, 2006

TD

Well it's a symmetrical matrix so it's surely diagonalizable.
I don't see any special 'trick' although the classical way (add $-\lambda$ on the main diagonal and compute the determinant) shouldn't be too hard thanks to the many 0's... (e.g. expand the determinant to the first row or column).

3. Jan 8, 2006

fargoth

yeah, im just REALLY lazy :tongue2:
i thought there's a trick that would make me see in a sec the right solution...
i only know the trick for block diagonal, and its not useful here...

4. Jan 8, 2006

TD

Perhaps there is, but I then I don't know it

If you know the (normal) method and you're only interested in the solution, why not use a computer program?

5. Jan 8, 2006

fargoth

well, the advantage of knowing helpful shortcuts is that you develop some intuition, which is pretty useful...

for example, if i got a block diagonal matrix i know the different blocks are orthogonal, which means vectors with parts that belong only to a certain block would stay in that block... and thats why i can find eigenvectors seperately for each block.

Last edited: Jan 8, 2006
6. Jan 8, 2006

TD

Of course, it was just a suggestion if you were only looking for the answer.
It's by far a better choice to do it yourself, if you wish to develop your mathematical intuition