       Re: orthonormal eigenvectors

• To: mathgroup at smc.vnet.net
• Subject: [mg88498] Re: orthonormal eigenvectors
• From: Szabolcs Horvát <szhorvat at gmail.com>
• Date: Tue, 6 May 2008 06:45:32 -0400 (EDT)
• Organization: University of Bergen
• References: <fvmmko\$86b\$1@smc.vnet.net>

```Bill Rowe wrote:
> 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:= m = {{1, -I (2^(1/2)), 0},
>     {I (2^(1/2)), 0, 0},
>     {0, 0, 2}};
>
> In:= ev = Eigenvectors[m]
>
> Out= {{0, 0, 1}, {-I Sqrt, 1, 0}, {I/Sqrt, 1, 0}}
>
> Then it is easy to normalize these by doing:
>
> In:= len = (#.#) & /@ ev;
> ev /Sqrt[len]
>
> Out= {{0, 0, 1}, {-Sqrt, -I, 0}, {I, Sqrt, 0}}
>
> But while all of these have unit length, they do not form an
> orthonormal set since
>
> In:= Dot @@ Rest[%]
>
> Out= -2 I Sqrt
>
> which is clearly not zero. That is for your matrix, a set of
> orthonormal eigenvectors doesn't exist.
>

The matrix is Hermitian, i.e. m == Conjugate@Transpose[m], so a set of
orthonormal eigenvectors is guaranteed to exist with respect to the dot
product complexDot = #1 . Conjugate[#2] &.

Even though one eigenvalue of the matrix is degenerate, the set of
vectors returned by Eigenvectors are orthogonal to each other, but not
normalized:

In:= ev = Eigenvectors[m]
Out= {{0, 0, 1}, {-I Sqrt, 1, 0}, {I/Sqrt, 1, 0}}

In:= Outer[complexDot, ev, ev, 1]
Out= {{1, 0, 0}, {0, 3, 0}, {0, 0, 3/2}}

Normalize[] handles complex vectors correctly, so we can simply do the
following:

In:= ev = Normalize /@ ev
Out= {{0, 0, 1}, {-I Sqrt[2/3], 1/Sqrt, 0}, {I/Sqrt,
Sqrt[2/3], 0}}

In:= Outer[complexDot, ev, ev, 1]
Out= {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}

```

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