RE: Generating Two Unit Orthogonal Vectors

*To*: mathgroup at smc.vnet.net*Subject*: [mg36448] RE: Generating Two Unit Orthogonal Vectors*From*: "Ersek, Ted R" <ErsekTR at navair.navy.mil>*Date*: Sat, 7 Sep 2002 02:54:04 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

David Park replied with ---------------- Daniel Lichtblau has pointed out that NullSpace does not generally give orthogonal vectors. Therefore the routines that depended upon that were in error. He says that it does give orthogonal vectors when the input vector contains approximate numbers. For graphical purposes this will be good enough for me. Therefore I modify Ted's routine to OrthogonalUnitVectors[vect__?(VectorQ[#, NumericQ] &)] /; (SameQ @@ Length /@ {vect}) && (Length[First[{vect}]] > 1) := #/Sqrt[#.#] & /@ NullSpace[{vect}// N] ---------------- Lets see what NullSpace does with approximate complex vectors. In[1]:= v1 = {1.0 I, 0.0, 0.5 I, 0.0, 1.0}; v2 = {0.0, 2.0, 1.0 I, 2.0, 0.5}; {v3,v4,v5} = NullSpace[{v1,v2}] Out[3]= {{-0.730153 + 0.*I, 0. - 0.138254*I, 0.250585 + 0.*I, 0. - 0.138254*I, 0. + 0.60486*I}, {0. + 0.*I, -0.515861 + 0.*I, 0. + 0.457321*I, 0.687357 + 0.*I, 0.22866 + 0.*I}, {0. + 0.*I, 0.510406 + 0.*I, 0. + 0.740442*I, -0.23274 + 0.*I, 0.370221 + 0.*I}} -------- In the next line we see NullSpace returned vectors that are orthogonal to the vectors we gave NullSpace. In[4]:= {v1.v3, v1.v4, v1.v5, v2.v3, v2.v4, v2.v5}//Chop Out[4]= {0, 0, 0, 0, 0, 0} ---------- However, the vectors returned aren't orthogonal to each other. In[5]:= {v3.v4, v3.v5, v4.v5}//Chop Out[5]= {0.229195*I, 0.371087*I, -0.677239} --------- I suppose an OrthogonalUnitVectors function that uses NullSpace should (1) Only accept real valued vectors. (2) Ensure NullSpace is given approximate vectors. ------ Regards, Ted Ersek