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Re: orthonormal eigenvectors

  • To: mathgroup at smc.vnet.net
  • Subject: [mg88415] Re: orthonormal eigenvectors
  • From: Bill Rowe <readnews at sbcglobal.net>
  • Date: Mon, 5 May 2008 06:10:40 -0400 (EDT)

On 5/3/08 at 6:18 AM, nairbdm at hotmail.com (. .) wrote:

>I have a 3X3 matrix M: {1,-i(2^(1/2)),0} {i(2^(1/2)),0,0} {0,0,2}

>And I am trying to find a set of orthonormal eigenvectors for M.

You can find a set of eigenvectors by doing:

In[54]:= m = {{1, -I (2^(1/2)), 0},
    {I (2^(1/2)), 0, 0},
    {0, 0, 2}};

In[55]:= ev = Eigenvectors[m]

Out[55]= {{0, 0, 1}, {-I Sqrt[2], 1, 0}, {I/Sqrt[2], 1, 0}}

Then it is easy to normalize these by doing:

In[56]:= len = (#.#) & /@ ev;
ev /Sqrt[len]

Out[57]= {{0, 0, 1}, {-Sqrt[2], -I, 0}, {I, Sqrt[2], 0}}

But while all of these have unit length, they do not form an
orthonormal set since

In[58]:= Dot @@ Rest[%]

Out[58]= -2 I Sqrt[2]

which is clearly not zero. That is for your matrix, a set of
orthonormal eigenvectors doesn't exist.


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