Right and Left Eigenvectors

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!

[Rengin]

%Thank you for all the answers! They were really helpful for me to
%understand that eigenvalue concept. I have one last question regarding this
%matter. My question starts with line 29
clear all
clc
A=[2 -1 2 ; 0 -1 4 ; 0 -2 5];
[V,D,W]=eig(A);
Lambda=eig(A);
% Right eigenvectors
v1=V(1:end,1);
v2=V(1:end,2);
v3=V(1:end,3);
% v4=V(1:end,4);
%Left eigenvectors
w1=W(1,1:end);
w2=W(2,1:end);
w3=W(3,1:end);
% w4=W(4,1:end);
%Diagonal values
d1=D(1,1);
d2=D(2,2);
d3=D(3,3);
% d4=D(4,4);
%Eigen values
e1=Lambda(1);
e2=Lambda(2);
e3=Lambda(3);
% e4=Lambda(4);
% From the explanation of eigenvectors
% A.vi=vi.ei-->correct for A.v(1...3)=v(1...3).e(1...3), right eig. vec.
% wi.A=ei.wi-->is not correct for wi(1...3).A=e(1...3).w(1...3), left eig. vec.
% where vi,wi and ei --> right,left eigenvectors and corresponding eigenvalue
% Why? Is there anything wrong in my understanding?
A*v1 == v1*e1; % for the values i=1,..3 , it is correct
w1*A == e1*w1; % for the values i=1,..3 , it is not correct.

[Steven Lord]

What relationship needs to hold between the left eigenvectors, the eigenvalues, and the original matrix? From the documentation: "[V,D,W] = eig(A) also returns full matrix W whose columns are the corresponding left eigenvectors, so that W'*A = D*W'." The columns of W are the left eigenvectors, not the rows.

>> W'*A-diag(Lambda)*W'

ans =

     1.0e-15 *

           0         0   -0.2220
           0         0         0
           0         0         0

Let's check with just one column of W, to more closely match your code.

>> w1c = W(:, 1);
>> w1c'*A - e1*w1c'

ans =

     1.0e-15 *

           0         0    0.2220

Those each look pretty good to me.

[Rengin]

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!

Answer 1

[Juan Fernandez]

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.

[Rengin]

Thank you for the answer Juan. Could you please take a look at my last edit to the question and help me understanding?

Answer 2

[Christine Tobler]

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)'

[Rengin]

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.