Quick Eigen Vector Question: Finding Eigen Vectors for Matrix A | Homework Help

In summary, Kamran is trying to solve a matrix equation but is having trouble finding the eigenvector for a given eigenvalue.
  • #1
17
0

Homework Statement



Hi, for the matrix A =
0 1 0

1 0 0

0 0 2

I have calculated the eigen values, and have successfully calculated the eigen vectors for lamda = -1 and 1. However for lamda = 2 the solutions says the answer is: Eigen vector: A( 0, 0, 1) where A is arbitrary and the vector is a column vector. The problem I am having is seeing where this answer came from.

Homework Equations



(A - lamda*I3)*(X, Y, Z) (column vector) = 0

The Attempt at a Solution


As I said above I already know the correct Eigen vector for this particular Eigen value, I just cannot see where the answer has come from. From using the equation above I got the three following equations:
-2X + Y = 0
-2Y = 0
0 = 0

From these equations I cannot see where you get an eigen vector including a value for Z. Any help would be very much appreciated, thanks in advance, Kamran.
 
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  • #2
superkam said:

Homework Statement



Hi, for the matrix A =
0 1 0

1 0 0

0 0 2

I have calculated the eigen values, and have successfully calculated the eigen vectors for lamda = -1 and 1. However for lamda = 2 the solutions says the answer is: Eigen vector: A( 0, 0, 1) where A is arbitrary and the vector is a column vector. The problem I am having is seeing where this answer came from.

Homework Equations



(A - lamda*I3)*(X, Y, Z) (column vector) = 0

The Attempt at a Solution


As I said above I already know the correct Eigen vector for this particular Eigen value, I just cannot see where the answer has come from. From using the equation above I got the three following equations:
-2X + Y = 0
-2Y = 0
0 = 0

From these equations I cannot see where you get an eigen vector including a value for Z. Any help would be very much appreciated, thanks in advance, Kamran.
The second equation tells you y=0, and then the first equation tells you x=0. There's no condition on z, so z is arbitrary. In other words, the eigenvector is of the form (0,0,t)=t(0,0,1).
 
  • #3
vela said:
The second equation tells you y=0, and then the first equation tells you x=0. There's no condition on z, so z is arbitrary. In other words, the eigenvector is of the form (0,0,t)=t(0,0,1).

Ok I understand now, thank you for your help. :)
 

1. What is an Eigen Vector?

An Eigen Vector is a vector that does not change direction when a linear transformation is applied to it. It is an important concept in linear algebra and is commonly used in various fields of science and mathematics.

2. How is an Eigen Vector different from a regular vector?

Unlike a regular vector, an Eigen Vector does not change direction when a linear transformation is applied to it. This means that it is a special type of vector that remains unchanged by certain mathematical operations.

3. What are some applications of Eigen Vectors?

Eigen Vectors have many practical applications, such as in image and signal processing, machine learning, quantum mechanics, and even in engineering and economics. They are also used in solving systems of linear equations and in finding patterns in data.

4. How do you find Eigen Vectors?

To find Eigen Vectors, you need to first calculate the Eigenvalues of a given matrix. Once you have the Eigenvalues, you can use them to find the corresponding Eigenvectors by solving a system of equations. Alternatively, you can also use software or programming languages like Python or MATLAB to find Eigen Vectors.

5. Why are Eigen Vectors important?

Eigen Vectors are important because they provide valuable insights into the behavior and characteristics of a system. They are used to identify patterns, reduce the dimensionality of data, and solve complex mathematical problems. In addition, they have many practical applications in various fields of science and engineering.

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