Re: Geometric series for matrices

*To*: mathgroup at smc.vnet.net*Subject*: [mg122369] Re: Geometric series for matrices*From*: "Dr. Wolfgang Hintze" <weh at snafu.de>*Date*: Wed, 26 Oct 2011 17:41:31 -0400 (EDT)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*References*: <j862is$5sl$1@smc.vnet.net>

"Evgeniya" <872dea at gmail.com> schrieb im Newsbeitrag news:j862is$5sl$1 at smc.vnet.net... > Hello, > I have a problem with thegeometric series for matrices. > > Suppose we define a matrix as > F = {{a1, a2}, {b1, b2}} > I need to find a sum of (wb)^i*F^i . One way I can do it is: > Fi = Sum[(w\[Beta])^i F^i, {i, 0, Infinity}] > It gives the answer: > Out={{1/(1 - a1 w\[Beta]), 1/(1 - a2 w\[Beta])}, {1/(1 - b1 > w\[Beta]), > 1/(1 - b2 w\[Beta])}} > > Another way is to use a formula (assuming convergence of course) > Inverse[{{1, 0}, {0, 1}} - w\[Beta] F] // FullSimplify > That gives the answer: > {{(1 - b2 w\[Beta])/(1 - w\[Beta] (b2 + a2 b1 w\[Beta]) + a1 w\[Beta] > (-1 + b2 w\[Beta])), -((a2 w\[Beta])/(-1 + w\[Beta] (a1 + b2 + a2 b1 > w\ > [Beta] - a1 b2 w\[Beta])))}, {-(( b1 w\[Beta])/(-1 + w\[Beta] (a1 + > b2 > + a2 b1 w\[Beta] - a1 b2 w\[Beta]))), (1 - a1 w\[Beta])/(1 - w\[Beta] > (b2 + a2 b1 w\[Beta]) + a1 w\[Beta] (-1 + b2 w\[Beta]))}} > > It seems thatMathematica uses the matrix of {{1,1},{1,1}} instead of > Identity. However I'm not sure exactly what to make out of it at this > point. I want to make sure I solve it correctly. > Thank you! > If you want normal matrix multiplication you have to use MatrixPower, otherwise F^k gives a matrix consisting of each element taken to the power k. Hence I would use InverseMatrix[MatrixPower[F,0]- w F] for the Sum[ MatrixPower[w F,k],{k,0,Infinity} ]. Note that MatrixPower[F,0] automatically gives you the unit matrix of the required dimension. --- Wolfgang