Eig command returning wrong eigenvectors?

Hi!

I am very new to using Matlab, so this question might seem silly.

I have a matrix:

    K =

      -7            -18            -18              9             27       
       0              3              0              0              0       
      -8            -12             -5              7             17       
     -10            -16            -20             10             25       
      -6            -10             -6              6             17

My goal is to calculate the eigenvectors and eigenvalues, which I attempt by using the command

    [M,N] = eig(A)

This command gives my two new matrices, one with the values and the other with the vectors. I get the right results for the values: 2, 3 and 5 (3 and 5 twice) but the vectors aren't correct:

M =
     273/1520         *            647/2153      -509/586          *       
       0              0              0              0            310/409   
    -273/1520       888/2737      2811/5162       266/1651       283/1695  
     273/304      -2220/2737      -847/1906      -452/1003      -293/7624  
    -273/760       1332/2737       813/1261      -446/3473       873/1387

The vectors should instead be something along the lines of (-1/2,0,1/2,-5/2,1).

What am I doing wrong? (I am assuming it's me and not the program)

Thanks in advance!

Answer 1

[John D'Errico]

Doing wrong? First, it looks like you have format set to rat.

[V,D] = eig(K);

V
V =
         0.179605302026751     -5.06266247219048e-14         0.300510985421823        -0.868600674748355      1.49364371455249e-15
                         0                         0                         0                         0         0.757946284651594
        -0.179605302026794         0.324442842261484         0.544556381601175         0.161114353924498         0.166961764835024
         0.898026510133868        -0.811107105653849        -0.444386053127199         -0.45064791217395       -0.0384312697617635
        -0.359210604053568         0.486664263392254         0.644726710075106        -0.128419204324953         0.629415789578334

diag(D)
ans =
          2.00000000000014
          2.99999999999972
          5.00000000000016
                         5
                         3

So eigenvalues of 2,3,5, the latter are of essentially multiplicity 2 each, if we ignore the trash in those least significant bits.

How about the eigenvectors?

You need to recognize that an eigenvector is unique only to within a scale factor. An eigenvector (V) basically has the property

K*V = lambda*V

for corresponding eigenvalue lambda.

But if that holds, then surely

K*(s*V) = lambda*(s*V)

for any scalar value s. So what happens if you scale those eigenvectors?

V(:,1)/V(1,1)
ans =
                       1
                       0
       -1.00000000000024
        5.00000000000064
       -2.00000000000037

How interesting.

So eig is NOT returning the wrong eigenvectors. It merely normalizes the eigenvectors so they have unit norm. Your expectations were wrong.

norm(V(:,1))
ans =
                       1

But that is just scaling the vectors by a constant. And we just discussed that that constant is completely arbitrary. At the same time, having unit normalized eigenvectors is very useful in mathematics.

[Lukas Esperi]

Thank you for a quick and thorough reply!