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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!
>
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