How to calculate eigen values and vectors of a matrix in matlab

Question

I have a square matrix C,of which I have to find the eigen values and eigen vectors.

C =

     2    -1    -1     0
    -1     3    -1    -1
    -1    -1     3    -1
     0    -1    -1     2

When I use the function eig() such that, [V,D]=eig(C,'nobalance') This is the output of V and D that I get:

V =

   -0.5000    0.7071    0.4914   -0.0924
   -0.5000   -0.0000   -0.3607    0.7874
   -0.5000   -0.0000   -0.6221   -0.6025
   -0.5000   -0.7071    0.4914   -0.0924

D =

   -0.0000         0         0         0
         0    2.0000         0         0
         0         0    4.0000         0
         0         0         0    4.0000

But when I give the same matrix C at an online matrix calculator(http://www.bluebit.gr/matrix-calculator/), the corresponding eigen values and vectors that I get are as follows:

Eigenvalues:

(0.000,0.000i)
(2.000,0.000i)
(4.000,0.000i)
(4.000,0.000i)

Eigenvectors:

( 0.500, 0.000i) ( 0.707, 0.000i) ( 0.500, 0.000i) (-0.308, 0.000i)
( 0.500, 0.000i) ( 0.000, 0.000i) (-0.500, 0.000i) (-0.250, 0.000i)
( 0.500, 0.000i) ( 0.000, 0.000i) (-0.500, 0.000i) ( 0.865, 0.000i)
( 0.500, 0.000i) (-0.707, 0.000i) ( 0.500, 0.000i) (-0.308, 0.000i)

Can you tell me why there is a difference in values at the two cases?Are the values returned by the eig function in matlab correct?Thanks in advance.

You can find out simply by trying to reconstruct the matrix from V and D... — Oliver Charlesworth
Incidentally, the matlab pages say that the balanceOption is ignored in cases where the input matrix is symmetric mathworks.co.uk/help/matlab/ref/eig.html#inputarg_balanceOption — mathematician1975
Yes,I could reconstruct the matrix C. But again, My project deals with eigen vectors calculated based on their signs. So when I gave the input C to the online calculater mentioned in the url, the signs of the figures are seen to be different. So wouldn't that be a problem? — Geethu Ann Joseph

Matt J Matt J · Accepted Answer · 2014-02-04T09:43:49

As you can see, the eigenvalues are the same. The eigenvectors corresponding to the eigenvalue 4 are different because that eigenvalue has multiplicity=2 and therefore its space of eigenvectors is two-dimensional. I.e., a numerical eigenvector solver could come up with any pair of linear independent vectors in that 2-dimensional space.

How to calculate eigen values and vectors of a matrix in matlab

1 Answers