Declaration of abstract matrices to perform series expansion on them

Question

I would like to have abstract matrices M and S to get out the coefficients of matrix power series however it treats M and S as numbers even if i checked that M.S - S.M != 0. I attach my code below:

$Assumptions = {Element[M, Matrices[{3, 3}]], Element[S, Matrices[{3, 3}]]}
Series[Simplify[(1 - a*(M + S))^(-1)], {a, 0, 3}]

How can i cure this to get out proper result?

Thanks in advance for reply.

Answer 1

Be explicit and do it in two steps. The first step is just the series computation with the matrix expression M+S replaced by a single variable:

f = Series[(1 + t x)^(-1), {t, 0, 3}]

$1-x t+x^2 t^2-x^3 t^3+O[t]^4$

We need to describe how to expand powers of x. This can be done recursively:

power[a_, n_Integer] /; n > 1 := Distribute[a . power[a, n - 1]];
power[a_, 1] := a;

After a power is expanded, we also need to collect sequences of like terms into (matrix) powers. This solution does it explicitly by post-processing the result of Split applied to such a sequence:

collect[a__] := With[{l = Power[First[#], Length[#]] & /@ Split[{a}]},
   If[Length@l == 1, First@l, Dot @@ l]];

Apply these rules to the series, remembering also to expand x itself when not wrapped inside Power:

f /. {Power[x, n_] :> power[m + s, n], x -> m + s} /. Dot[a__] :> Dot[collect[a]]

$(1+(-m-s) t+(m.m+m.s+s.m+s.s) t^2+ \\ (-m.m.m-m.m.s-m.s.m-m.s.s-s.m.m-s.m.s-s.s.m-s.s.s) t^3+ \\ O[t]^4)$

For some reason the second replacement does not work (MMA 8.0). It appears to work with almost any head except Dot! Here's a workaround:

f /. {Power[x, n_] :> power[m + s, n], x -> m + s} /.  x_[a__] /; x === Dot :> x[collect[a]]

$1+(-m-s) t+(m^2+s^2+m.s+s.m) t^2+ \\ (-m^3-s^3-m.s^2-m^2.s-s.m^2-s^2.m-m.s.m-s.m.s) t^3+O[t]^4$

Answer 2

There is a new function as of version 9 :

Series[MatrixFunction[(1 - a*(#))^(-1) &, bigM + bigS], {a, 0, 3}] 

While the output does not look very pretty it will behave correctly.

Answer 3

Is

$$\begin{split} 1+&(M+S) \, a\\ +&(M\cdot S+S\cdot M+M\cdot M+S\cdot S) \, a^2\\ +&(M\cdot M\cdot S+M\cdot S\cdot M+M\cdot S\cdot S\\ &+S\cdot M\cdot M+S\cdot M\cdot S+S\cdot S\cdot M+M\cdot M\cdot M+S\cdot S\cdot S) \, a^3\\ +&O\left(a^4\right) \end{split}$$

what you're expecting for? If yes, please continue read.

First we define a temporary matrix $A$:

A /: Times[A, A] := CircleTimes[A, A]
A /: Power[A, n_ /; IntegerQ[n] && Positive[n]] := CircleTimes @@ ConstantArray[A, n]
A /: Times[A, CircleTimes[M__]] := CircleTimes[A, M]
A /: Times[CircleTimes[M__], A] := CircleTimes[M, A]

Then we substitute $M+S$ with $A$ and expand the series as normal way:

tempSeries = Series[1/(1 - a  A), {a, 0, 3}]

$1+A\, a+A\otimes A\, a^2+A\otimes A\otimes A\,a^3+O\left(a^4\right)$

Then substitute $M+S$ back and do some replacement and transformation like Distribute etc.

tempSeries /. A -> M + S /.
  CircleTimes[M__] :> Distribute[CircleTimes[M]] /.
 CircleTimes -> Dot

Now we get the result shown at beginning.