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Suppose I have a rectangular matrix $M(x)$ depending on a real parameter $x$. I want to find out for which $x$ are the columns of $M(x)$ linearly dependent (or equivalently, when $M(x)$ has a nontrivial kernel). I thought about using NullSpace as in

NullSpace[M[x]]

but it stubbornly outputs an empty vector, I don't think it's able to tell me when there are nontrivial solutions. If for some reason I know of a solution $x_0$, NullSpace[M[x0]] outputs the correct nullspace, but how do I find $x_0$? If $M$ was square I could simply solve $\mathrm {Det}(M(x))=0$, but I don't have such luck.

How can I do this?

user2723984
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  • You can try to do a numerical search, otherwise you need a symbolic representation of your nullspace that can be solved for solutions – b3m2a1 Apr 09 '20 at 16:50
  • @b3m2a1 Hi, thanks for the comment, I'm only interested in analytical solutions unless it's impossible. What do you mean by symbolic representation of the NullSpace? Is there a way to have a symbolic representation beyond $M(x)=0$? The NullSpace might have different dimensions for various values of $x$. – user2723984 Apr 09 '20 at 16:58
  • I mean that it seems from the code like you want to know the null space in general for any arbitrary value of x, which is equivalent to having an analytic form for it. You might want to approach this from some other perspective at least to get some insights, say via the pseudo-inverse or something. – b3m2a1 Apr 09 '20 at 17:32
  • This might help. Also I think I have seen another related question and will try to find it. – Daniel Lichtblau Apr 09 '20 at 18:11
  • This one shows a general method. Augment on the right with an identity matrix. Row reduce. Check for denominators. Where they vanish gives necessary conditions for drops in the matrix rank. – Daniel Lichtblau Apr 09 '20 at 18:20
  • @DanielLichtblau hi! Thanks! Your comment seems quite interesting, but I don't understand what you mean. The link you've given lets you do row reduction showing steps, how does it help me? If I try to simply to RowReduce on my matrix it doesn't work (behaves similarly to NullSpace) – user2723984 Apr 10 '20 at 08:31
  • (1) Recall I stated you should augment with an identity matrix. Did you do that prior to using RowReduce? (2) If you edit the post to provide an explicit example, I might be able to demonstrate what I had in mind. – Daniel Lichtblau Apr 10 '20 at 14:27

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