Re: Geometric series for matrices
- To: mathgroup at smc.vnet.net
- Subject: [mg122360] Re: Geometric series for matrices
- From: Heike Gramberg <heike.gramberg at gmail.com>
- Date: Wed, 26 Oct 2011 17:39:53 -0400 (EDT)
- Delivered-to: l-mathgroup@mail-archive0.wolfram.com
- References: <201110251017.GAA05846@smc.vnet.net>
Power (^) has attribute Listable which means that {{a,b},{c,d}}^k is expanded to {{a^k,b^k},{c^k,d^k}} which is probably not what you want. To take the j-th power of a matrix F you should use MatrixPower instead, e.g. Fi = Sum[(w\[Beta])^i MatrixPower[F, i], {i, 0, Infinity}] // FullSimplify which gives as you second method. Heike On 25 Oct 2011, at 12:17, Evgeniya wrote: > 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! >
- References:
- Geometric series for matrices
- From: Evgeniya <872dea@gmail.com>
- Geometric series for matrices