Eigenvectors and using them in matrix algebra.

In summary, the conversation discusses the use of eigenvectors to solve A^11 and the concept of diagonalization. By using eigenvectors and diagonalization, A^11 can be simplified to a much easier calculation.
  • #1
Zhiv
5
0
Hi.

Marix A=

|1 1 0 |
|0 2 0 |
|2 1-1 |

Has three eigenvectors [1,1,1]^T, [1,0,1]^T and [0,0,1]^T, By using this knowledge solve A^11.

Ok, solving A^11 is rather easy with any decent calculator, or even with pen , paper and some time, but how on Earth I'm supposed to benefit from thoose eigenvectors?

Thank you.
 
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  • #2
Look up "diagonalization".

Let P be the 3x3 matrix whose columns consists of the eigenvectors of A. Then P^-1AP is a diagonal matrix (where the entries are the eigenvalues of the eigenvectors). If D is that diagonal matrix, we have P^-1AP = D <=> A = PDP^-1, so that A^m = P * D^m * P^-1 for natural m (the last step can be proven with induction). But calculating D^m is easy, just raise each non-zero entry to the power of m. Then it's just a matter of working out what P^-1 is, and then multiplying the matrices.
 
Last edited:
  • #3


Eigenvectors are a powerful tool in matrix algebra because they allow us to simplify complex matrix operations. In this case, we can use the eigenvectors to easily calculate A^11.

First, we need to find the eigenvalues corresponding to each eigenvector. We can do this by solving the characteristic equation det(A-λI)=0. In this case, we get eigenvalues of λ=1,2,-1.

Next, we can use the diagonalization theorem to write A as A=PDP^-1, where P is a matrix with the eigenvectors as its columns and D is a diagonal matrix with the eigenvalues on the diagonal.

So, we have A=[1,1,0; 1,0,1; 0,0,1] [1,0,0; 0,2,0; 0,0,-1] [1,-1,0; -1,2,0; 0,0,1].

Now, we can easily calculate A^11 by simply raising the diagonal matrix D to the 11th power, which gives us [1,0,0; 0,2048,0; 0,0,-1].

Finally, we can use P and P^-1 to transform the result back to the original basis, giving us A^11=[2048,2048,0; 0,4096,0; 4096,2048,-1].

As you can see, by using eigenvectors and diagonalization, we were able to simplify the calculation of A^11. This is just one example of how eigenvectors can be used in matrix algebra to make complex operations more manageable. They also have many other applications in fields such as engineering, physics, and computer science. So, while it may seem like a simple calculation in this case, eigenvectors are a valuable tool to have in your toolbox when working with matrices.
 

1. What are eigenvectors and why are they important in matrix algebra?

Eigenvectors are special vectors that do not change direction when multiplied by a specific matrix. They are important in matrix algebra because they help us understand the behavior and properties of matrices, as well as simplify calculations and solve complex problems.

2. How do you find eigenvectors of a matrix?

To find eigenvectors of a matrix, you need to first find the eigenvalues of the matrix. This can be done by solving the characteristic equation of the matrix. Once you have the eigenvalues, you can then use them to find the corresponding eigenvectors by solving a system of linear equations.

3. What is the significance of the eigenvalue-eigenvector relationship?

The eigenvalue-eigenvector relationship is significant because it allows us to decompose a matrix into simpler components, making it easier to understand and manipulate. It also helps us identify important patterns and structures within a matrix.

4. Can eigenvectors be negative?

Yes, eigenvectors can have negative values. This is because eigenvectors only represent the direction of a vector, not its magnitude. Therefore, the direction of an eigenvector can be positive or negative, depending on the values of the matrix it is associated with.

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

Eigenvectors are used in a variety of real-world applications such as image and signal processing, data analysis, and machine learning. They are also used in physics, engineering, and economics to model and understand complex systems. Additionally, they have applications in cryptography and network analysis.

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