# Find Eigenvalues & Eigenvectors for Exercise 3 (2), Explained!

• member 731016
In summary, the conversation discusses finding eigenvectors and solving a matrix equation using simple operations. The solution involves finding the determinant of the matrix and solving for possible values of lambda. The conversation also touches on understanding row reduced echelon form and using operations such as subtracting rows and dividing rows to manipulate the matrix. However, it is noted that special treatment is needed in the case where a equals -1.
member 731016
Homework Statement
Relevant Equations
For exercise 3 (2),
,
The solution for finding the eigenvector is,

However, I am very confused how they got from the first matrix on the left to the one below and what allows them to do that. Can someone please explain in simple terms what happened here?

Many Thanks!

You are looking for vectors for which ##Ax=\lambda x##. This is solvable if the determinant of ##A-\lambda I## is zero. Should give you three possible values for ##\lambda## , (different or two or three equal). You then solve ##(A-\lambda I)\vec x=\vec 0## with simple operations like subtracting one row from another.

##\ ##

member 731016
ChiralSuperfields said:

For exercise 3 (2),
View attachment 327486,
The solution for finding the eigenvector is,
View attachment 327487
However, I am very confused how they got from the first matrix on the left to the one below and what allows them to do that. Can someone please explain in simple terms what happened here?

Many Thanks!
When you apply the concept as required; you shall end up with the equation,

##(-1-λ)(λ^2-aλ+λ-a)=0##.

you picked your ##λ=-1## and if you substitute the eigen value back into the matrix, you can confirm that it satisfies the equation.

...

from there, we now have to now deal with the equation, ##(2+2a)x+(1+a)z=0##

Last edited:
SammyS and member 731016
BvU said:
You are looking for vectors for which ##Ax=\lambda x##. This is solvable if the determinant of ##A-\lambda I## is zero. Should give you three possible values for ##\lambda## , (different or two or three equal). You then solve ##(A-\lambda I)\vec x=\vec 0## with simple operations like subtracting one row from another.

##\ ##

Sorry what operations were used to get from,

Orange matrix to pink matrix?

Many thanks!

chwala said:
When you apply the concept as required; you shall end up with the equation,

##(-1-λ)(λ^2-aλ+λ-a)=0##.

you picked your ##λ=-1## and if you substitute the eigen value back into the matrix, you shall end up with,

still posting

ChiralSuperfields said:

Sorry what operations were used to get from,
View attachment 327490
Orange matrix to pink matrix?

Many thanks!
...switching of rows combined with understanding of row reduced echelon form...

member 731016
ChiralSuperfields said:
what operations were used to get from orange matrix to pink matrix?
Subtract row 2 from row 3 (gives ##0,0,0##)
Divide row 2 by ##2+2a## (gives ##1,0,{1\over 2}##)
Swap row 1 and 2

##\ ##

member 731016 and chwala
BvU said:
Subtract row 2 from row 3 (gives ##0,0,0##)
Divide row 2 by ##2+2a## (gives ##1,0,{1\over 2}##)
Swap row 1 and 2

##\ ##

Note that dividing a row by $2 + 2a$ is only valid if $a \neq -1$. This case will require separate treatment.

chwala and member 731016

## What are eigenvalues and eigenvectors?

Eigenvalues are scalars that indicate how much the corresponding eigenvectors are stretched or compressed during a linear transformation. Eigenvectors are non-zero vectors that change only in scale (not direction) when a linear transformation is applied.

## How do you find the eigenvalues of a matrix?

To find the eigenvalues of a matrix, you need to solve the characteristic equation, which is obtained by subtracting λ times the identity matrix from the original matrix and setting the determinant to zero: det(A - λI) = 0. The solutions to this equation are the eigenvalues.

## How do you find the eigenvectors once you have the eigenvalues?

Once the eigenvalues are found, the eigenvectors can be determined by substituting each eigenvalue back into the equation (A - λI)v = 0, where A is the original matrix, λ is the eigenvalue, I is the identity matrix, and v is the eigenvector. Solve this system of linear equations to find the eigenvectors.

## What is the significance of eigenvalues and eigenvectors in practical applications?

Eigenvalues and eigenvectors have significant applications in various fields such as physics, engineering, computer science, and economics. They are used in stability analysis, vibration analysis, facial recognition, principal component analysis (PCA) in statistics, and solving differential equations, among others.

## Can a matrix have complex eigenvalues and eigenvectors?

Yes, a matrix can have complex eigenvalues and eigenvectors, especially if the matrix is not symmetric. Complex eigenvalues often occur in systems exhibiting oscillatory behavior or in certain transformations in the complex plane.

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