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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