Order of Eigenvectors in Diagonalization

In summary, When solving a problem about diagonalization, the speaker knows how to find eigenvalues and eigenvectors, but struggles with ordering them. They mention that in their book, the largest value is often placed first, but in another example, the order is 0, 1, 1. The speaker acknowledges that there is not one specific way to diagonalize a matrix and suggests choosing an ordering that works best for the individual. They apologize for any confusion and mention having a test in the near future.
  • #1
blackrose75
5
0
I essentially know how to find eigenvalues and thus eigenvectors, though when solving a problem about diagonalization I do not know how to order them (as in, I can find all the eigenvectors but do not know which order to place them into find my X that diagonalizes my A)

In the examples of my book it first seems to be the largest value (putting the eigenvector corresponding to the eigenvector 1 first then the one to -4), but when I go to another example it puts them in order of 0, 1, 1 (each being one of the eigenvalues.)

Apologies if I phrased this question confusingly, it's a bit late and my test is in another day or two.

Thanks.
 
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  • #2
there's not "one way" to diagonalize a matrix, there's several ways, corresponding to different orderings of the eigenvalues. pick an order of the eigenvalues that works for you.
 

What is the "Order of Eigenvectors" in Diagonalization?

The "Order of Eigenvectors" in Diagonalization refers to the arrangement of the eigenvectors in the diagonal matrix used to represent a linear transformation. The eigenvectors are placed in the same order as the corresponding eigenvalues.

How are the Eigenvectors arranged in Diagonalization?

In Diagonalization, the eigenvectors are arranged in a diagonal matrix, with each eigenvector occupying a different row or column. The matrix is ordered in such a way that the eigenvectors correspond to their respective eigenvalues.

Why is it important to consider the Order of Eigenvectors in Diagonalization?

The Order of Eigenvectors in Diagonalization is important because it determines the transformation of a vector under a linear transformation. The arrangement of the eigenvectors in the diagonal matrix dictates the direction and magnitude of the transformation on a given vector.

Can the Order of Eigenvectors be changed in Diagonalization?

No, the Order of Eigenvectors cannot be changed in Diagonalization. The eigenvectors must be arranged in the same order as the corresponding eigenvalues in order for the diagonalization process to be valid.

How does the Order of Eigenvectors affect the resulting diagonal matrix in Diagonalization?

The Order of Eigenvectors determines the specific values in the diagonal matrix, and therefore, affects the entire matrix. Changing the order of eigenvectors would result in a different diagonal matrix and thus, a different representation of the linear transformation.

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