# Solving equations for Eigenvector: Vanishing

• zak100
In summary, the two equations are equivalent and you can solve either one for the solution to the system.
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.

zak100 said:

## 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.

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

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

#### Attachments

• Solving Decimal eq for Eigen vectors LDA vanishing_PicOfSlide.jpg
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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##.

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}##

Zulfi.

## 1. What does it mean for an Eigenvector to vanish?

When an Eigenvector vanishes, it means that its corresponding Eigenvalue is equal to zero. This means that when the Eigenvector is multiplied by its respective matrix, the resulting vector will be a zero vector.

## 2. Why is solving equations for Eigenvector vanishing important?

Solving equations for Eigenvector vanishing is important because it allows us to find the special Eigenvectors that correspond to a matrix's zero Eigenvalues. These Eigenvectors play a critical role in understanding the behavior of a matrix and its transformations.

## 3. How do you solve equations for Eigenvector vanishing?

To solve equations for Eigenvector vanishing, we need to find the Eigenvectors that correspond to a matrix's zero Eigenvalues. This can be done by setting up and solving a system of equations where the variables are the entries of the Eigenvector.

## 4. Can a matrix have more than one Eigenvector that vanishes?

Yes, a matrix can have multiple Eigenvectors that correspond to a zero Eigenvalue. In fact, if a matrix has a zero Eigenvalue, it will have an infinite number of Eigenvectors that correspond to it.

## 5. What is the significance of the Eigenvectors that vanish for a given matrix?

The Eigenvectors that vanish for a given matrix are significant because they represent the special directions or vectors that are not changed by the matrix's transformation. These Eigenvectors form the basis for understanding the behavior of a matrix and its underlying structure.

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