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Re: RE: RE: Generating Two Unit Orthogonal Vectors

• To: mathgroup at smc.vnet.net
• Subject: [mg36493] Re: [mg36476] RE: [mg36448] RE: Generating Two Unit Orthogonal Vectors
• From: Daniel Lichtblau <danl at wolfram.com>
• Date: Mon, 9 Sep 2002 00:29:35 -0400 (EDT)
• References: <200209080731.DAA05434@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

DrBob wrote:
> 
> Wouldn't it be nice if NullSpace's behavior were DOCUMENTED?  Otherwise,
> it's futile to give it approximate numbers expecting any particular
> behavior.  Even if it always works, it may not work in the next version
> of Mathematica.
> 
> Bobby

The expected, and documented, behavior is that the output should be a
basis for the null space, that is, solutions of the homogeneous matrix
equation A.x==0. If this were to stop working then that would be a
serious bug. Is this the behavior you mean?

The implementation notes of the manual mention that approximate
NullSpace is based on a singular values decomposition. This in fact
gives resulting vectors that are orthonormal by the usual
conjugate-symmetric inner product on C (though these are now not
"normal" to the original vector in this same inner product, unless they
are real-valued). But this basis-orthogonality is not part of the
mission of NullSpace and moreover should not become part of it. Hence
that particular (and implementation dependent) aspect of NullSpace
should not become documented.

Daniel Lichtblau
Wolfram Research

• References:
  • RE: RE: Generating Two Unit Orthogonal Vectors
    • From: "DrBob" <drbob@bigfoot.com>