[SOLVED] infinite products of matrices

**choovuck** · August 13th 2008, 07:04 AM

hey all,

the basic fact about infinite products of complex numbers is that $\prod_{j=1}^\infty a_j$ is convergent if and only if $\sum_{j=1}^\infty |1-a_j|$ converges. does anyone know if something of this kind holds for matrix-valued case?

I hope for $\prod_{j=1}^\infty A_j$ converges if $\sum_{j=1}^\infty ||A_j-I||<\infty$ .

an easier related statement, which looks very true (but I still don't know solution yet) is to prove $\prod_{j=1}^\infty e^{B_j}$ converges if $\sum_{j=1}^\infty ||B_j||<\infty$ (here $B_j$ 's are not assumed to commute, of course)

if anyone saw this matter addressed in any book or science paper, I'd appreciate the reference

**Opalg** · August 13th 2008, 11:39 AM

Originally Posted by choovuck

I hope for $\prod_{j=1}^\infty A_j$ converges if $\sum_{j=1}^\infty ||A_j-I||<\infty$ .

I prefer to write I+A_j in place of A_j, and to assume that $M \mathrel{\mathop=^{\text{d}\text{ef}}} \sum_{j=1}^\infty ||A_j||<\infty$ .

Let $T_n = \prod_{j=1}^n (I+A_j)$ . Then $\|T_n\|\leqslant \prod_{j=1}^n \|I+A_j\| \leqslant \prod_{j=1}^\infty (1+\|A_j\|) \leqslant e^M,$ by the usual estimate for infinite products.

Also, if m < n then $\|T_m-T_n\| = \left\|T_m\left(I-\prod_{j=m+1}^n (I+A_j)\right)\right\| \leqslant e^M\left\|I-\prod_{j=m+1}^n (I+A_j)\right\|.$ But $\left\|I-\prod_{j=m+1}^n (I+A_j)\right\|$ should lie between $1-e^{-K}$ and $e^K-1$ , where $K = \sum_{j=m+1}^n\|A_j\|.$ That makes it look to me as though (T_n) should be a Cauchy sequence, which would therefore converge. Of course, you won't want the infinite product to diverge to 0, so you would also need to investigate whether $\|T_n\|$ is bounded away from 0 (presumably by $e^{-M}$ ).

I don't pretend to have thought through this carefully, so I can't be sure that there is any mileage in this approach. Maybe you have already tried it. But it looks reasonably promising to me.

**choovuck** · August 13th 2008, 12:35 PM

this seems to be working, thanks!

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