Right and Left Eigenvectors

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Rengin
Rengin on 25 Jun 2018
Commented: Rengin on 27 Jun 2018
Dear Users,
What I read about right and left eigenvectors from some literature is that right ones should be column and left ones should be row matrices. However, when I use the standard commands ([V,D,W]=eig(A)) to see right(V), diagonal(D) and left(W) vectors of e.g 3x3 square matrix, I get 3x3 V and W matrices. Can you tell me where I do a mistake? Thanks!
  3 Comments
Rengin
Rengin on 27 Jun 2018
Dear Steven Lord, "The columns of W are the left eigenvectors, not the rows." is the sentence I was looking for. Unfortunately, the book I have says that left eigenvectors are row vectors even though Matlab documentation says other way around. Therefore, I was confused. Thank you for your time!

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Answers (2)

Juan Fernandez
Juan Fernandez on 25 Jun 2018
You did not make a mistake. Each column of V is a right eigenvector of A. Each row of W is a left eigenvector of A.
  1 Comment
Rengin
Rengin on 27 Jun 2018
Thank you for the answer Juan. Could you please take a look at my last edit to the question and help me understanding?

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Christine Tobler
Christine Tobler on 25 Jun 2018
The matrices V and W contain all eigenvectors of A. For example
V(:, 1) and D(1, 1) are the right eigenvector and eigenvalue of A, A*V(:, 1) == V(:, 1)*D(1, 1)
W(:, 1) and D(1, 1) are the left eigenvector and eigenvalue of A, W(:, 1)'*A == D(1, 1)*W(:, 1)'
  1 Comment
Rengin
Rengin on 27 Jun 2018
Dear Christine. Thank you very much for your detailed explanation. I have still a small understanding problem. I would really appreciate if you can take a look at my last editing to the related question.

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