Find the eigenvalues and eigenvectors

[Mutatis]

Homework Statement

Find the eigenvalues and eigenvectors fro the matrix: $$
A=\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} $$.

Homework Equations

Characteristic polynomial: $\nabla \left( t \right) = t^2 - tr\left( A \right)t + \left| A \right|$ .

The Attempt at a Solution

I've found the eigenvalues doing through the characteristic polynomial equation above: $$ \lambda_1 = 1 $$ $$ \lambda_2 = -1 $$.
Then, to get the eigenvector associated to $\lambda_1$ the equation $M v_1 = 0$ must be satisfied, $$ \begin{pmatrix} \left(0-1\right) & -i \\ i & \left(0-1\right) \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = 0$$.
It leads me to a system that I'm having trouble to solve it: $$ \begin{cases} -x-iy=0 \\ ix-y=0 \end{cases} $$.
I don't know what to do next, please help me!

 

[dRic2]

Why are you having trouble ? What are you expecting to find ?

 

[RPinPA]

As I'm sure you realized, these equations are linearly dependent so the system has infinitely many solutions. So parametrize it, let $y$ be anything. Say $y = a$. Solve for $x$. That's a valid eigenvector. Any multiple of that is a valid eigenvector.

 

[Mutatis]

parametrize it, let yyy be anything. Say y=ay=ay = a. Solve for xxx. That's a valid eigenvector. Any multiple of that is a

I did what you've said $y=1$ then $x=1/i=-i$, so I got $v_1=(1, -i)$. When I put this vector in the matrix to verify $Mv_1=0$ it leads me to a non-zero value...

 

[Ray Vickson]

I did what you've said $y=1$ then $x=1/i=-i$, so I got $v_1=(1, -i)$. When I put this vector in the matrix to verify $Mv_1=0$ it leads me to a non-zero value...

That's odd: when I put $x = -i$ and $y = 1$ into $-x - iy$ I get $0$, exactly as wanted. The second left-hand-side is just $i \times$ the first left-hand-side, so it will equal $0$ also.

 

[RPinPA]

I did what you've said $y=1$ then $x=1/i=−i$, so I got $v_1=(1,−i)$.

Are you sure? If $x$ is $-i$ and $y$ is $1$, you would write that as $(1, -i)$?

 

[Mutatis]

Oh! Guys I'm sorry I wrote the values wrong! Now I understand what I was doing wrong. Thank you very much guys!