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What is meant by generic rank of a matrix? Is it something different from the rank, and does the word generic has just its English meaning? I came across this term in the book "Algebraic statistics for biology" (ed Lior Pachter and Bernd Sturmfels) theorem 19.5.

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    Is there any particular reason why you're not quoting the sentence in which you encountered this expression? And you don't by any chance mean *Algebraic Statistics for Computational Biology*? Theorem 19.5 can be viewed at [Google Books](http://books.google.com/books?id=zjeVawvaA5QC&lpg=PP1&dq=Algebraic%20statistics%20for%20computational%20biology&hl=de&pg=PA349#v=onepage&q=generic%20rank&f=false).2012-04-15
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    I didn't think the sentence would provide any additional information that would help anyone in answering the question ("The generic rank of the associated matrix is ..."). And yes, that's the book I am talking about.2012-04-15

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I don't have the book, but I'll make a guess: I suspect the matrix in question depends on one or more parameters, and the author means that for "generic" values of those parameters the matrix has a certain rank. In this context "generic" can mean "in a dense $G_\delta$ set". For example, the matrix $$\pmatrix{p & 0\cr 0 & q\cr}$$ has rank $2$ unless $p=0$ or $q=0$, so you might say it has generic rank $2$.