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? >