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## 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!