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MathGroup Archive 2006

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Re: schur decomposition and mathematica

  • To: mathgroup at smc.vnet.net
  • Subject: [mg66993] Re: schur decomposition and mathematica
  • From: "Jeremy Watts" <jwatts1970 at hotmail.com>
  • Date: Tue, 6 Jun 2006 06:28:08 -0400 (EDT)
  • References: <e5tqf3$cvk$1@smc.vnet.net> <e60nvv$g53$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

<bghiggins at ucdavis.edu> wrote in message news:e60nvv$g53$1 at smc.vnet.net...
> Jeremy,
>
> That is correct. If the eigenvalues are complex, they "appear" in 2X2
> blocks along the diagonal of T in the decomposition A=Q.T.Q* . In your
> case the eigenvalues are complex
>
> mat={{-5.0, 7.0, 6.0, -3.0}, {2.0, -8.0, 3.0, 9.0}, \
> {7.0, 1.0, 2.0, -7.0}, {3.0, 4.0, 4.0, 4.0}};
>
> Eigenvalues[mat]
>
> {-9.80269 + 0.336644 I, -9.80269 - 0.336644 I,
>  6.30269 + 4.81699 I, 6.30269 - 4.81699 I}
>
>  Now consider the Schur decomposition of mat
>
> {Q, T} = SchurDecomposition[mat]
>
> The T matrix is
>
> T
>
> {{6.30269,-6.10921,1.83689,5.59659},{3.79811,6.30269,
>
> 1.7072,6.28934},{0.,0.,-9.80269,6.03753},{0.,0.,-0.0187708,-9.80269}}
>
> Thus in the first 2x2 block the diagonal values are the real part
> (6.30269) and the off-diagonal terms can be used to construct the
> imaginary part as follows:
>
>
> Sqrt[T[[1,2]]*T[[2,1]]]
>
> 0. + 4.8169937117972985*I
>
>
> Hope this helps,
>
> Cheers,
>
> Brian

thanks brian,

i thought that that might be the case, but technically i thought T had to be
strictly upper triangular, which in this case it is not.  i did write my own
schur decomposition program recently and went through all sorts of problems
trying to 'iron out' these 2x2 buldges (which i did eventually), but maybe
now i shouldnt have bothered... who am i to argue with steven wolfram :)
>
>
> {Q, T} = SchurDecomposition[]
> Jeremy Watts wrote:
> > why upon entering :-
> >
> > SchurDecomposition[{{-5.0, 7.0, 6.0, -3.0}, {2.0, -8.0, 3.0, 9.0}, {7.0,
> > 1.0, 2.0, -7.0}, {3.0, 4.0, 4.0, 4.0}}]
> > does mathematica return a 'T matrix' that is not upper triangular?
> >
> > some the eigenvalues are complex, so is it because these eigenvalues
reside
> > along the diagonal of 'T' in 2x2 blocks?
>


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