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I have the following matrix;

matrix={{0.213052 - 5.52399*10^-18 I, 0.123451 + 0.215784 I, 
  0.0771636 + 0.298479 I}, {0.123451 - 0.215784 I, 
  0.328615 - 4.47928*10^-18 I, 
  0.367929 + 0.0925207 I}, {0.0771636 - 0.298479 I, 
  0.367929 - 0.0925207 I, 0.458333 - 8.30535*10^-18 I}}

when I found Eigenvectors[matrix] in Mathematica 9, it gave the following output;

{{0.117433 + 0.4424 I, 0.552311 + 0.14137 I, 
  0.682245 + 0. I}, {-0.131023 - 0.625986 I, 
  0.730765 + 0. I, -0.163119 + 0.174212 I}, {0.010945 - 
   0.617533 I, -0.372767 + 0.0446533 I, 0.691075 + 0. I}}

But when I found Eigenvectors[matrix] in Mathematica 10, it gave different output which is;

{{-0.117433 - 0.4424 I, -0.552311 - 0.14137 I, -0.682245 + 
   0. I}, {-0.367396 + 0.523493 I, -0.499467 - 0.533433 I, 
  0.238658 + 0. I}, {0.010945 - 0.617533 I, -0.372767 + 0.0446533 I, 
  0.691075 + 0. I}}

After all my calculations the out put obtained from Mathematica 9 gives correct final result.

How i can understand this problem? And how to handle it? Thanks.

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1 Answer 1

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The eigenvectors are defined up to an arbitrary complex constant. In other words, if $v_i$ is an eigenvector then $c_iv_i$ is the same eigenvector. Usually eigenvectors is normalized so $|c_i|=1$.

v9 = {{0.117433 + 0.4424 I, 0.552311 + 0.14137 I, 
    0.682245 + 0. I}, {-0.131023 - 0.625986 I, 
    0.730765 + 0. I, -0.163119 + 0.174212 I}, {0.010945 - 
     0.617533 I, -0.372767 + 0.0446533 I, 0.691075 + 0. I}};
v10 = {{-0.117433 - 0.4424 I, -0.552311 - 0.14137 I, -0.682245 + 
     0. I}, {-0.367396 + 0.523493 I, -0.499467 - 0.533433 I, 
    0.238658 + 0. I}, {0.010945 - 0.617533 I, -0.372767 + 0.0446533 I,
     0.691075 + 0. I}};

Chop[v10/v9]
(* {{-1., -1., -1.}, {-0.683483 - 0.729965 I, -0.683485 - 
   0.729965 I, -0.683484 - 0.729965 I}, {1., 1., 1.}} *)

Abs[%]
(* {{1., 1., 1.}, {0.999999, 1., 1.}, {1., 1., 1.}} *)

So the eigenvectors are the same (0.999999 is due to small precision of the copy-paste)

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  • $\begingroup$ ybeltukov thanks. But in a lengthy code i desire the result of v9 while using mathematica 10. What is your kind advice on it? $\endgroup$
    – Usman
    Commented Jan 21, 2015 at 16:11
  • $\begingroup$ It doesnt make any difference to the solution $\endgroup$
    – Philipp
    Commented Jan 21, 2015 at 16:23

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