Matrix element of a field

Hi, I am not 100% sure my working is correct for this question. Could someone be kind enough to check my working please.

The characteristic polynomial of A:

So the eigenvalues are and .

To find the eigenvectors we sub in and into the equations:

to solve for x and y for both eigenvalues.

So for example, for I get and .

Am I correct in my working out so far?

Thanks

Re: Matrix element of a field

Re: Matrix element of a field

Quote:

Originally Posted by

**FernandoRevilla** I suppose

. Then, the characteristic polynomial of

is

so, the eigenvalues of

are

or equivalently

.

Oh yes, 3 isn't an element of , thanks for that.

Why is -i equivalent to 2i? I know it is a stupid question.

Then how do I work out the eigenvectors?

Thanks

Re: Matrix element of a field

Re: Matrix element of a field

Quote:

Originally Posted by

**FernandoRevilla** For example, if

then

As

is simple,

. Solving you'll find a basis, for example

.

Thanks FernandoRevilla, but I don't really understand this. Could you please explain?

I can't see how to get the eigenvectors.

Thanks, and sorry for being a pain.

Re: Matrix element of a field

Quote:

Originally Posted by

**Juneu436** Thanks FernandoRevilla, but I don't really understand this. Could you please explain? I can't see how to get the eigenvectors.

Could you specify your doubts?. These things are just routine knowing the corresponding theory.

Re: Matrix element of a field

Quote:

Originally Posted by

**FernandoRevilla** Could you specify your doubts?. These things are just routine knowing the corresponding theory.

This section: Quote:

As

is simple,

. Solving you'll find a basis, for example

.

How does and how do I go from there to find the eigenvectors?

Thanks again for your help.

Re: Matrix element of a field

Re: Matrix element of a field

or you could just solve the system Av = λv for each eigenvalue λ.

for example, with λ = i, and writing a+bi = z, c+di = w this leads to:

that is:

so:

so any eigenvector corresponding to i is of the form (z,(1+i)z)...z, of course, must be non-zero. it is convenient to choose z = 1+0i = 1.