Finding eigenvalues and eigenvectors

[fargoth]

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

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 that's because i know what this matrix is... but if i'll see some matrix with different values then this roaming around... i don't know what i'll do, i don't 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!-

 

[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).

 

[fargoth]

yeah, I am just REALLY lazy
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...

 

[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?

 

[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 that's why i can find eigenvectors separately for each block.

 

[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