Eigenvectors & normalising

[kel]

Homework Statement

Hi, I'm going over an old exam paper as part of my revision for upcoming exams (joy ! ok, maybe not :yuck: )

Anyway, I've gotten myself a bit lost here and would appreciate some guidance.
Here we go:

Compute the eigenvalues & eigenvectors of the following matrix. Normalise the 2 eigenvectors.

Matrix = (3 1+i)
(1-i 2)

Homework Equations

The Attempt at a Solution

So far I have the 2 eigenvalues being +1 and +4, but I'm having trouble with the next bit, I think that the eigenvectors will come out as complex numbers,

so far I have reduced to two equations:

2x + (1+i)y = 0
(1-i)x + y = 0

ie 2x + (y+iy) = 0 (so, 2x = -y-iy)
and x-ix + y = 0 (and y = -x+ix)

Is this correct?? if not why? and where do I go from here?

Also, I don't get the idea of normalisation (next part of question), I understand that it has something to do with setting the vector(i think) to 1, but I'm not sure. My lecturers notes aren't the easiest to follow outside of his lectures, so if you could walk me through it a bit I'd be very grateful.

Thanks
Kel

 

[cristo]

Homework Statement

Hi, I'm going over an old exam paper as part of my revision for upcoming exams (joy ! ok, maybe not :yuck: )

Anyway, I've gotten myself a bit lost here and would appreciate some guidance.
Here we go:

Compute the eigenvalues & eigenvectors of the following matrix. Normalise the 2 eigenvectors.

Matrix = (3 1+i)
(1-i 2)

Homework Equations

The Attempt at a Solution

So far I have the 2 eigenvalues being +1 and +4, but I'm having trouble with the next bit, I think that the eigenvectors will come out as complex numbers,

Is this a problem? You are working in a complex vector space, and so the eigen-vectors are entitled to have complex entries!

so far I have reduced to two equations:

2x + (1+i)y = 0
(1-i)x + y = 0

ie 2x + (y+iy) = 0 (1) (so, 2x = -y-iy)
and x-ix + y = 0 (2) (and y = -x+ix)

Is this correct?? if not why? and where do I go from here?

I've not checked your work, but you look to be doing the correct method. Why not try solving eqns (1) and (2) for x and y?

Also, I don't get the idea of normalisation (next part of question), I understand that it has something to do with setting the vector(i think) to 1, but I'm not sure. My lecturers notes aren't the easiest to follow outside of his lectures, so if you could walk me through it a bit I'd be very grateful.

Normalised means that the vectors have unit length. Once you have the eigenvector, v, say, then the normalised vector is v/|v|