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May 15th, 2016, 08:59 AM   #1
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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
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May 22nd, 2016, 02:57 AM   #2
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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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