Eigenvectors and using them in matrix algebra.

[Zhiv]

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.

 

[Muzza]

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.

 

[M98Ranger]

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.