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MathGroup Archive 2011

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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!



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