How Do You Find Eigenvectors of a 2x2 Matrix?

In summary, to find the eigenvalues and eigenvectors of a matrix, one must first find the determinant of the matrix minus the identity matrix multiplied by the eigenvalue, which will result in a quadratic equation. The roots of this equation will be the eigenvalues of the matrix. Then, using one of the eigenvalues, one can find the corresponding eigenvector by setting one of the entries of the vector equal to a multiple of the other. This process should be repeated for the other eigenvalue to find its corresponding eigenvector. Finally, one should verify their results by using the matrix and the calculated eigenvectors to confirm that they satisfy the equation Ax = λx, where λ is the corresponding eigenvalue.
  • #1
zak100
462
11

Homework Statement


Consider the following Matrix:

Row1 = 2 2

Row2 = 5 -1
Find its Eigen Vectors

Homework Equations


Ax = λx & det(A − λI)= 0.

The Attempt at a Solution


First find the det(A − λI)= 0. which gives a quadratic eq.
roots are
λ1 = -3 and λ2 = 4 (Eigen values)

Then using λ1, I got:

x1= -2/5 x2

Now how can i find out the vectors?

Some body please guide me.

Zulfi.
 
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  • #2
The eigenvectors must be normalized, no?
 
  • #3
Hi,
I don't know about that. I am attaching the example which I have followed. It has not discussed about Normalization.

Zulfi
 

Attachments

  • Eigen vectors of 2 x 2 matrix.docx
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  • #4
zak100 said:

Homework Statement


Consider the following Matrix:

Row1 = 2 2

Row2 = 5 -1
Find its Eigen Vectors

Homework Equations


Ax = λx & det(A − λI)= 0.

The Attempt at a Solution


First find the det(A − λI)= 0. which gives a quadratic eq.
roots are
λ1 = -3 and λ2 = 4 (Eigen values)

Then using λ1, I got:

x1= -2/5 x2
And x2 = x2 (meaning that x2 is arbitrary).

I haven't checked your work, but if it's correct, any eigenvector for ##\lambda = 3## is a scalar multiple of <-2/5, 1>.
zak100 said:
Now how can i find out the vectors?

Some body please guide me.

Zulfi.
 
  • #5
Can you post what it says? I cannot open the file.
 
  • #6
Hi,
You are not correct. It says: -2 5. I don't know what would be the vector if:
x1 = -2x2.
Will it be : -2 1
It says about my question:
This means that, while there are infinitely many nonzero solutions (solution vectors) of the
equation Ax = −3x, they all satisfy the condition that the first entry x1 is −2/5 times the
second entry x2.
[ 2t ] = t [2 ]
−5t -5

(Note my square bracket is small. It should enclose 2t & -5t and the other one should enclose 2 -5)
,

where t is any real number. The nonzero vectors x that satisfy Ax = −3x are called
eigenvectors associated with the eigenvalue = −3. One such eigenvector is
u1 =[ 2 −5]

Some body please guide .

Zulfi.
 
  • #7
zak100 said:
Hi,
You are not correct.
Who are you replying to? Use the Quote feature to copy what someone else is saying.
zak100 said:
It says: -2 5. I don't know what would be the vector if:
x1 = -2x2.
Will it be : -2 1
It says about my question:
This means that, while there are infinitely many nonzero solutions (solution vectors) of the
equation Ax = −3x, they all satisfy the condition that the first entry x1 is −2/5 times the
second entry x2.
[ 2t ] = t [2 ]
−5t -5

(Note my square bracket is small. It should enclose 2t & -5t and the other one should enclose 2 -5)
,

where t is any real number. The nonzero vectors x that satisfy Ax = −3x are called
eigenvectors associated with the eigenvalue = −3. One such eigenvector is
u1 =[ 2 −5]
That looks good. To summarize what you have, if one eigenvalue is ##\lambda = -3##, its associated eigenvector is ##\vec x = \begin{bmatrix} 2 \\ -5\end{bmatrix}##. You can and should verify this fact by confirming that ##A\begin{bmatrix} 2 \\ -5\end{bmatrix} = -3 \begin{bmatrix} 2 \\ -5\end{bmatrix}##, using the matrix you wrote in post #1.

Now do the same kind of work to find the other eigenvalue and associated eigenvector, and you're done (but you should check this one as well).
 

Related to How Do You Find Eigenvectors of a 2x2 Matrix?

1. What is an eigenvector?

An eigenvector is a vector that does not change direction when a linear transformation is applied to it. In other words, when a matrix is multiplied by an eigenvector, the resulting vector is a scalar multiple of the original eigenvector.

2. How do I find eigenvectors for a 2*2 matrix?

To find the eigenvectors for a 2*2 matrix, you will need to find the eigenvalues first. You can do this by solving the characteristic equation, which is obtained by setting the determinant of the matrix minus the identity matrix equal to 0. Once you have the eigenvalues, you can plug them back into the original matrix to solve for the corresponding eigenvectors.

3. What is the significance of eigenvectors in linear algebra?

Eigenvectors are important in linear algebra because they provide a way to understand how a linear transformation affects a vector. They also help to simplify calculations, as it is often easier to work with eigenvectors than with the original matrix.

4. Can a matrix have more than two eigenvectors?

Yes, a matrix can have any number of eigenvectors. However, for a 2*2 matrix, there can be at most two linearly independent eigenvectors.

5. How are eigenvectors used in real-life applications?

Eigenvectors have many real-life applications, such as in physics, engineering, and economics. They are used to study the behavior of complex systems, make predictions, and solve differential equations. For example, in physics, eigenvectors are used to find the normal modes of a vibrating system, and in economics, they are used to analyze the stability of a financial system.

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