Eigenvalue Help!

I have been accepted into a MSc. Economics program and am now trying to fill out the gaps in my math before I start.

I am, at the moment, trying to teach myself Eigenvalues and Eigenvectors, however I can't seem to work it out for a 3x3 matrix.

I am obviously doing something wrong as I can't construct the characteristic equation of the 3x3 and the book I am using just skips that part (as do the notes from the University) so I imagine its something pretty simple that I am missing.

I have attached the example and all help would be greatly appreciated.

Cokehead Mother

Deleted User wrote: »

normally end up with a cubic equation instead of (5 - x)(x -4)(2 - x) = 0

Could you post the cubic you ended up with? You definitely should end up with one, which you can then factorise to find the eigenvalues.

If you know that (5 - x)(x -4)(2 - x) = 0 is what you should have then you can just multiply out the brackets to see if the your cubic is right, and that you found the determinent correctly.

Its the product of:

(1-x)(5-x)(2-x)-6(5-x)

Which isn't adding up to me but I did work it out on the back of an envelope. Will be heading into the 'the office' soon and will try it there again.

Cokehead Mother

That looks just fine.

If you clean it up you get (x - 4)(x - 5)(x + 1) = 0 which gives you eigenvalues 4, 5 and -1.

If you sub these values back into your matrix, the determinent will equal 0.

The equation (5 - x)(x - 4)(2 - x) = 0 gives you eigenvalues 4, 5 and 2. But when you sub lambda = 2 back into the matrix, the determinant of the matrix doesn't equal 0 so (5 - x)(x - 4)(2 - x) = 0 can't be right.

Thanks - was really stressing over that last night!