# Power series with matrix coefficients

Let $A(t)\in SL_r(\mathbb C((t)))$ be a formal power series with matrix coefficients, and let $B(t)\in SL_r(\mathbb C[t])$ and $C(t)\in SL_r(\mathbb C[[t]])$ such that : $$A(t)=B(1/t) \;( ^tA(-t)^{-1})C(t)$$ This implies in particular $$A(t)=B(1/t) \;( ^tB(-1/t)^{-1})A(t)(^tC(-t)^{-1})C(t)$$

Question:

• Is it true that we must have $(^tC(-t)^{-1})C(t)=1$ and $B(1/t) \;( ^tB(-1/t)^{-1})=1$?

• What can we say about $C(t)$ and $B(t)$? In particular, what is their degree and order respectively?

• Is there a description of $A^{-1}(t)$? for example, what is its order? and is some coefficients must be invertible?

Any comment or suggest are welcomed

Thanks.

• I think you need some more assumptions to have an affirmative the answer. What if $A=0$, or what if e.g. $B= C^{-1}$ are constant matrices and commute with $A$? – Pietro Majer Jun 3 '15 at 21:30
• for sure , A is not 0. and I don't know about the other – Z.A.Z.Z Jun 3 '15 at 21:46
• that's what I'm saying: in the given assumptions one can't exclude these simple cases, for which the answer is clearly negative. – Pietro Majer Jun 4 '15 at 21:59
• I can assume that $B$ does not commute with $A$ and add the equation $$A(t)=B(1/t) (^tA(-t)^{-1})C(t)$$, using the order, I get $B(t)=B_0$ (does not depend in $t$), could I get another information? or could one construct a interesting example? – Z.A.Z.Z Jun 7 '15 at 9:02