not getting left eigen vector

Hi,

I have attached the matlab code file

syms z1 z2 z3 z4 z5 z6 z7 ;

A=[z1 z2 z3; z4 z5 z6; 1 2 z7]

M=subs(A,[z1 z2 z3 z4 z5 z6 z7],[-0.1 0.1 -2 0.01 0.07 -0.5 0.2])

[V,D] = eig(M)

[W, ~] = eig(M.')

W'*M - D*W'

The W'*M-D*W' should be equal to or almost zero but that not the case in my coding.

The W'*M-D*W' outputs are like this

ans =

1.0e-14 *

-0.1388 0.5329 0

ans =

1.9774 + 0.1985i 4.0123 - 0.0596i 0.0000 + 3.4528i .(this should be equal to zero)

ans =

1.9774 - 0.1985i 4.0123 + 0.0596i 0.0000 - 3.4528i. (This should be equal to zero)

Is there any different command which I can use for correct left eigen vectors or need any modification

in this command. Please help

Answer 1

[John D'Errico]

If you want the set of left and right eigenvectors, we see in the help for eig...

[V,D,W] = eig(A) also produces a full matrix W whose columns are the

corresponding left eigenvectors so that W'*A = D*W'.

Note this is not the case for the symbolic version of eig. Oh well. But you are converting everything to a double in the end. So why in the name of god and little green apples did you need to start with a symbolic form, and then immediately substiture floating point numbers for all variables?

M = [-0.1,  0.1, -2;...
     0.01 , 0.07, -0.5;...
     1,  2,  0.2];

[V,D,W] = eig(M)

V = 
   0.7416 + 0.0000i   0.7416 + 0.0000i   0.8954 + 0.0000i
   0.1860 - 0.0199i   0.1860 + 0.0199i  -0.4413 + 0.0000i
  -0.0625 - 0.6412i  -0.0625 + 0.6412i  -0.0591 + 0.0000i

D = 
   0.0936 + 1.7264i   0.0000 + 0.0000i   0.0000 + 0.0000i
   0.0000 + 0.0000i   0.0936 - 1.7264i   0.0000 + 0.0000i
   0.0000 + 0.0000i   0.0000 + 0.0000i  -0.0173 + 0.0000i

W = 
   0.3491 - 0.0403i   0.3491 + 0.0403i   0.2455 + 0.0000i
   0.7097 + 0.0000i   0.7097 + 0.0000i  -0.9689 + 0.0000i
  -0.0091 - 0.6106i  -0.0091 + 0.6106i   0.0300 + 0.0000i

norm(M*V - V*D)

ans = 1.9481e-15

So zero to within floating point trash. How about the other set of eigenvectors? Eig states this form should hold for the left eigenvectors.

W'*A = D*W'

Does it?

norm(W'*M - D*W')

ans = 9.5794e-16

Again, zero, to within floating point trash.

So what did you do wrong? You computed RIGHT eigenvectors of M'. (M is real, so the .' is irrelevant.) This time I will do it in symbolic form, using the sym/eig code to compute the left eigenvectors.

M
M =
[-1/10,  1/10,   -2]
[1/100, 7/100, -1/2]
[    1,     2,  1/5]

[W,D] = eig(M');

So we must have this be true, for the RIGHT eigenvectors of M'.

M' * W - W * D

But in the form of left eigenvectors of M', we can transpose that, to get

W' * M - D' * W'

Lets try it out.

vpa(W' * M - D' * W',16)
ans =
[                        -2.522900868286734e-23,                          1.025703959565754e-22, 0]
[- 2.9778502051909e-23 + 3.639594695233322e-23i, 9.926167350636332e-23 - 2.827406731282818e-22i, 0]
[- 2.9778502051909e-23 - 3.639594695233322e-23i, 9.926167350636332e-23 + 2.827406731282818e-22i, 0]

Do you see that is zero, again to within floating point trash?

norm(W' * M - D' * W')
ans =
0

What did you do wrong? You assumed that D' was equal to D.

Is D complex? YES. Was your assumption correct? NO.

[Abhinav Kumar]

Thank you very much for your answer Sir.

Actually, I am estimating the symbols using optimization technique. I substituted the values midway in order to check whether I am going in right direction or not of finding the eigen vectors. It is true by your help.Thank you again.