Matrix decomposition with NullSpace and QRDecomposit

*To*: mathgroup at smc.vnet.net*Subject*: [mg64139] Matrix decomposition with NullSpace and QRDecomposit*From*: Tobias Burnus <burnus at gmx.de>*Date*: Thu, 2 Feb 2006 00:06:45 -0500 (EST)*Sender*: owner-wri-mathgroup at wolfram.com

(Retried - this time w/o attachment; forget to de-attach it before sending the mail - sorry.) Hello, I have a Hermitian matrix of the form ( A B ) ( B' C ) =: M (B' = Transpose[B]), of which I want to bring B (5×18) into this form {D, O} = d11 d12 d13 d14 d15 0 0 0 0 0 0 0 0 0 0 0 0 0 d21 d22 d23 d24 d25 0 0 0 0 0 0 0 0 0 0 0 0 0 d31 d32 d33 d34 d35 0 0 0 0 0 0 0 0 0 0 0 0 0 d41 d42 d43 d44 d45 0 0 0 0 0 0 0 0 0 0 0 0 0 d51 d52 d53 d54 d55 0 0 0 0 0 0 0 0 0 0 0 0 0 where O should be zero. For a simple case I succeeded using Q = QRDecomposition[B] L = NullSpace[Transpose[B]], (see below) but it fails for a more complicated case - there not all (O)ij are zero. (See http://www.physik.fu-berlin.de/~tburnus/tmp/MatrixDecomposition.nb ) This is with Mathematica 5.2. It works if I set pds1=pds2=...=pds6 and pdp1=...pdp6. Any ideas? Besides, does anyone know a quick way to get D Hermitian, currently it has triangular form. Tobias Excerpt from the notebook -------- HC1 = (.... the matrix ...) HC1C := HC1[[Range[1, 5], Range[6, 23]]] Q = QRDecomposition[ Transpose[HC1C]][[1]] L = NullSpace[HC1C] T = IdentityMatrix[Length[HC1]]; T[[Range[6, 6 + Length[Q] - 1], Range[6, Length[HC1]]]] = Q; \!\(\(T\[LeftDoubleBracket]Range[6 + Length[Q], Length[HC1]], Range[6, Length[HC1]]\[RightDoubleBracket]\ = \ Table[L\[LeftDoubleBracket]i\[RightDoubleBracket]\/\@\(\(Conjugate[L]\)\ \[LeftDoubleBracket]i\[RightDoubleBracket] . L\[LeftDoubleBracket]i\ \[RightDoubleBracket]\), {i, 1, Length[L]}];\)\) T=Assuming[ pd\[Sigma]1<0&&pd\[Pi]1>0&&pd\[Sigma]2<0&&pd\[Pi]2>0&&pd\[Sigma]3<0&& pd\[Pi]3>0&&pd\[Sigma]4<0&&pd\[Pi]4>0&&pd\[Sigma]5<0&&pd\[Pi]5>0&& pd\[Sigma]6<0&&pd\[Pi]6>0,FullSimplify[T]]; myHC1 = T.HC1.Transpose[T] // FullSimplify

**Follow-Ups**:**Re: Matrix decomposition with NullSpace and QRDecomposit***From:*Pratik Desai <pdesai1@umbc.edu>