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Hello everyone,

Are there any known conditions to ensure that the singular value of a matrix A is smaller than 1 ? More specifically, in my case A is the product of an M-Matrix and an inverse M-Matrix, if it can help in any way. I haven't found anything yet.

Thank you for your help

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    $\begingroup$ Do you mean all the singular values, I assume? $\endgroup$
    – Federico Poloni
    Commented Jan 16, 2012 at 18:30
  • $\begingroup$ If elements are quite small, then sing.vals. will be small, it can be made precise... but is what you are looking for ? $\endgroup$
    – Alexander Chervov
    Commented Jan 16, 2012 at 18:45
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    $\begingroup$ I am seeing votes to close, but no comments with reasons. At any rate, if you don't feel like explaining what an M-matrix is, you might have better luck on scicomp.stackexchange.com $\endgroup$
    – S. Carnahan ♦
    Commented Jan 17, 2012 at 3:28
  • $\begingroup$ Sorry for being imprecise. I am interested in the largest singular value only. An M-Matrix has mainly two properties : 1- all non diagonal entries are negative 2- all principal minors are positive A sufficient condition to check that a matrix is an M-Matrix is that it can be written (\lambda I - B) for some nonnegative matrix B and some \lambda > p, where p is a maximal eigenvalue of B. A property of an M-Matrix is that all the eigenvalues are positive. I don't know if these properties can help. But yes, I am trying to find conditions to ensure that the largest s.v. is < 1. Thanks, $\endgroup$
    – user20638
    Commented Jan 17, 2012 at 20:38
  • $\begingroup$ To Alexander Chervov (or anyone else who knows), You say that "If elements are quite small, then sing.vals. will be small, it can be made precise". How can it "be made precise" ? I would be interested to know more (I am not myself mathematician). In my case, the elements of A are all smaller that between -1 and 1. Don't know if it helps... Thank you. $\endgroup$
    – user20638
    Commented Jan 20, 2012 at 21:56

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