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.