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.