My Math Forum Relationship between a rank and the limit of a matrix

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 May 15th, 2016, 07:59 AM #1 Newbie   Joined: May 2016 From: London Posts: 1 Thanks: 0 Relationship between a rank and the limit of a matrix So I have a question, Let's suppose I have a matrix $M$, where all the elements $a_{ij}$ are linear expressions that are functions of an arbitrary number of variables i.e. $$a_{ij}=f_{ij}(x_{1},x_{2},...,x_{n})$$ Let's suppose that I take the limit of one of these variables to infinity i.e. Take $x_{1}\rightarrow\infty$ and call the resulting matrix $M_{x_{1}\rightarrow\infty}$ My question is are the ranks of these two matrices related? Put in other words Is $rank(M)$ related to $rank(M_{x_{1}\rightarrow\infty})$ Does anyone know the answer to this question or know of a text/source that does have the answer? Thank you
 May 22nd, 2016, 01:57 AM #2 Senior Member   Joined: Aug 2012 Posts: 229 Thanks: 3 Hey AdemH. If your function is indeed a linear one then the matrix itself should remain static. You usually apply an operator (including a matrix) to a vector of values and evaluate it. The matrix and its properties are usually static and therefore don't change even if the mapping does as a function of its inputs.

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