Can you help me find Eigenvectors for this matrix?

[M55ikael]

Homework Statement

Hi! I have this frustrating equation I can't solve even though I have the answer.
I'm trying to make two Eigenvectors for my matrix. I have found λ1 and λ2, but I can't solve the resulting equations.

Homework Equations

A = \begin{bmatrix} 2 & 1 \\ -\frac{1}{2} & 1 \end{bmatrix}


λ1= 3/2 + j/2 , λ2= 3/2 - j/2

\begin{bmatrix} 2 -(3/2 + j/2) & 1 \\ -\frac{1}{2} & 1 - (3/2 + j/2) \end{bmatrix} * E1 = 0



Attempt at solution:

E1
α(-1-j) + 2β = 0
-α + β(-1-j) = 0


E2
α(-1+j) + 2β = 0
-α + β(-1+j) = 0

I can't make any sense out of those equations though.

Answers should be
E1 = \begin{bmatrix} 2 \\ -1+j \end{bmatrix} * c

E2 = \begin{bmatrix} 2 \\ -1-j \end{bmatrix} * t

c and t are different from 0

 

[Mark44]

Homework Statement

Hi! I have this frustrating equation I can't solve even though I have the answer.
I'm trying to make two Eigenvectors for my matrix. I have found λ1 and λ2, but I can't solve the resulting equations.

Homework Equations

A = \begin{bmatrix} 2 & 1 \\ -\frac{1}{2} & 1 \end{bmatrix}


λ1= 3/2 + j/2 , λ2= 3/2 - j/2

\begin{bmatrix} 2 -(3/2 + j/2) & 1 \\ -\frac{1}{2} & 1 - (3/2 + j/2) \end{bmatrix} * E1 = 0

This looks fine. Maybe you got fouled up in the arithmetic. The matrix can be simplified a bit, which will make the subsequent work easier.

\begin{bmatrix} 1/2 - i/2 & 1 \\ -1/2 & -1/2 - i/2 \end{bmatrix}

I'm using i for the imaginary unit...

This row reduces to
\begin{bmatrix} 1 & 1 + i \\ 0 & 0 \end{bmatrix}

The top row says that x₁ = -(1 + i)x₂, so an eigenvector is
\begin{bmatrix} -1 - i \\ 1 \end{bmatrix}

Attempt at solution:

E1
α(-1-j) + 2β = 0
-α + β(-1-j) = 0


E2
α(-1+j) + 2β = 0
-α + β(-1+j) = 0

I can't make any sense out of those equations though.

Answers should be
E1 = \begin{bmatrix} 2 \\ -1+j \end{bmatrix} * c

E2 = \begin{bmatrix} 2 \\ -1-j \end{bmatrix} * t

c and t are different from 0

 

[M55ikael]

Thanks, I allways knew it was pretty simple. But it's so easy to get your pluses and minuses mixed up, I couldn't get it reduced properly.