Geometric series for matrices

*To*: mathgroup at smc.vnet.net*Subject*: [mg122325] Geometric series for matrices*From*: Evgeniya <872dea at gmail.com>*Date*: Tue, 25 Oct 2011 06:17:32 -0400 (EDT)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com

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!

**Follow-Ups**:**Re: Geometric series for matrices***From:*Heike Gramberg <heike.gramberg@gmail.com>