Solving equations for Eigenvector: Vanishing

[zak100]

Homework Statement

I am trying to solve a Eigen vector matrix:

$\begin{bmatrix}9.2196& 6.488\\4.233& 2.9787\end{bmatrix}\cdot\begin{bmatrix}x\\y\end{bmatrix}-\lambda\begin{bmatrix}x\\y \end{bmatrix}=0$

I have found $\lambda_1 = 0$ and $\lambda_2 = 12.1983$
However, I can't solve the following equations:
$9.2196x + 6.488y =0 -------(eq.1)$
and $4.233x + 2.978y =0 ----(eq.2)$

I am multiplying $(eq.1)$ by $4.233x$ and $(eq.2)$ by $9.2196$

Some body please guide me how to solve these equations for $x$ and $y$ values.

Zulfi.

Homework Equations

$9.2196x + 6.488y =0 -------(eq.1)$
and $4.233x + 2.978y =0 ----(eq.2)$​

The Attempt at a Solution

I am multiplying $(eq.1)$ by $4.233$ and $(eq.2)$ by $9.2196$ but both the equations are vanishing. Some body please guide me.

Zulfi.

 

[PeroK]

Homework Statement

I am trying to solve a Eigen vector matrix:

$\begin{bmatrix}9.2196& 6.488\\4.233& 2.9787\end{bmatrix}\cdot\begin{bmatrix}x\\y\end{bmatrix}-\lambda\begin{bmatrix}x\\y \end{bmatrix}=0$

I have found $\lambda_1 = 0$ and $\lambda_2 = 12.1983$
However, I can't solve the following equations:
$9.2196x + 6.488y =0 -------(eq.1)$
and $4.233x + 2.978y =0 ----(eq.2)$

I am multiplying $(eq.1)$ by $4.233x$ and $(eq.2)$ by $9.2196$

Some body please guide me how to solve these equations for $x$ and $y$ values.

Zulfi.

Homework Equations

$9.2196x + 6.488y =0 -------(eq.1)$
and $4.233x + 2.978y =0 ----(eq.2)$​

The Attempt at a Solution

I am multiplying $(eq.1)$ by $4.233$ and $(eq.2)$ by $9.2196$ but both the equations are vanishing. Some body please guide me.

Zulfi.

The two equations are equivalent. So, you just take the solution of either one.

 

[hilbert2]

And, it only determines the solution up to a constant multiplier, so you can for example arbitrarily set $x=1$ and then find $y$.

 

[zak100]

Hi,
Thanks for your response. Answer is different in the slide. I have attached the slide. Please guide me.

 

[hilbert2]

In the solution for $w_1$, the ratio of the components $\frac{0.8178}{0.5755}$ is the same as the ratio $\frac{9.2196}{6.488}$ in your system of equations. So it's the same answer but with different normalization, one where $x^2 + y^2 = 1$.

 

[zak100]

Hi,
$lambda_2 =12.1983$
I am not able to get the correct answer.

$9.2196x + 6.488y = 12.1983x$
$9.216x - 12.1983x = -6.488y$

$\begin{bmatrix}x\\y\end {bmatrix} = \begin{bmatrix}2.7\\1\end{bmatrix}$

Somebody please guide me how can i get the correct answer?
Zulfi.