Solving for complex eigenvalues

[trap101]

Quick question:

I have a characteristic polynomial: λ² + i = 0...how do I solve for the eigenvalues?

They're suppose to be + or - (√2/2)(1 - i) How'd they get those?

 

[Dick]

Quick question:

I have a characteristic polynomial: λ² + i = 0...how do I solve for the eigenvalues?

They're suppose to be + or - (√2/2)(1 - i) How'd they get those?

λ^2=(-i)=exp(i*3*pi/2). Use deMoivre.

 

[trap101]

I'm kind of shaky with deMoivre, is there another way?

 

[Dick]

I'm kind of shaky with deMoivre, is there another way?

Only harder ways. deMoivre is the easy way. One square root of exp(i*t) is exp(i*t/2). What does that lead to your case? Then use exp(i*t/2)=cos(t/2)+i*sin(t/2).

 

[trap101]

Only reason I ask is because we were not really shown the deMoivre way in class,...to think of it, we weren't even shown a way to solve for these complex roots. I'll give deMoivre's a try.

Another question, just out of curiosity:

Say I have a basis for the Ker(T). T being the transformation. Say I have an arbitrary vector v in V. Can I write v as a linear combination of the basis of the Ker(T), and if so do the co-efficients that I obtain from this linear combination make a co-ordinate vector in V, that if I apply the transformation T to this co-ordinate vector, will it map to the null-space in Im(T)?

 

[Dick]

Only reason I ask is because we were not really shown the deMoivre way in class,...to think of it, we weren't even shown a way to solve for these complex roots. I'll give deMoivre's a try.

Another question, just out of curiosity:

Say I have a basis for the Ker(T). T being the transformation. Say I have an arbitrary vector v in V. Can I write v as a linear combination of the basis of the Ker(T), and if so do the co-efficients that I obtain from this linear combination make a co-ordinate vector in V, that if I apply the transformation T to this co-ordinate vector, will it map to the null-space in Im(T)?

Ok, for the first one you need (a+bi)(a+bi)=a^2-b^2+2abi=(-i). Equating real and imaginary parts you get two equations for the real numbers a and b. Now try to solve them.

For the second one. You can't write an arbitrary element of element of V as a combination of the basis of Ker(T) unless it's in Ker(T). And then it will map to 0 in Im(T). I'm not sure what the real question is here?

 

[trap101]

For the second one. You can't write an arbitrary element of element of V as a combination of the basis of Ker(T) unless it's in Ker(T). And then it will map to 0 in Im(T). I'm not sure what the real question is here?

Yup, that's what I was asking about.

Ok, for the first one you need (a+bi)(a+bi)=a^2-b^2+2abi=(-i). Equating real and imaginary parts you get two equations for the real numbers a and b. Now try to solve them.

so I get two equations: a²-b² = 0 and 0 + 2abi = -i Is that what you mean?

I'm also trying the deMoivre way. So as of now, I'm converting -i into polar co-ordinates in order to use de Moivre: so currently I have 1 sin$\sigma$, how do you get $\sigma$ to be pi/2 ?

 

[trap101]

well I figured out the pi/2, how it's obtained; but I got -pi/2. I got it from b = r sin$\sigma$ ==>

-1 = 1 sin$\sigma$, and then figured out for $\sigma$.

 

[trap101]

wanted to delete this message...I'm actually still puzzled when it comes to the -pi/2

 

[Dick]

wanted to delete this message...I'm actually still puzzled when it comes to the -pi/2

exp(-i*pi/2)=cos(-pi/2)+i*sin(-pi/2)=(-i), right? A square root of that is exp(-i*pi/4)=cos(-pi/4)+i*sin(-pi/4). Which equals?