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Math Help - [SOLVED] infinite products of matrices

  1. #1
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    Unhappy [SOLVED] infinite products of matrices

    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
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  2. #2
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    Quote Originally Posted by choovuck View Post
    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.
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    this seems to be working, thanks!
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