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Finding conjugating matrix

Posted 4 months ago
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Hi, I'm completely new to Mathematica, and trying to do a quick crash course to get what I need to be able to perform some computations for part of a research problem. I've currently got two matrices $X$ and $C$ which I know to be conjugate, and I'm trying to get Mathematica to return the matrix (of a particular form, unipotent) that conjugates them. Here's a minimal working example:

X={{a[1],1},{b[1],-a[1]}}
C={{0,1},{b[1]-a[1]*a[1],0}}
L={{1,0},{l[2,1],1}}

I'm trying to get Mathematica to solve the equation X.L==L.C for the entries of L below the diagonal, and, in particular, return the matrix L that solves this equation. It's probably worth noting that, I'd need to be able to generalise this to larger matrices ( $X$ and $C$ always known $n$-by-$n$ matrices, but are written in terms of variables a[1],...,a[n-1],b[1],...,b[n-1]), and L will have $(n^2-n)/2$ entries to be solved for.

A little background on what I've found, but doesn't seem to work:

  • I tried "Thread" for the equation, and solved. But this returns "solutions" for every entry involved. I believe the needed solutions (l[i,j], with i>j) are amongst the solutions, but I couldn't find how to extract these solutions. Maybe if that can be done, then Mathematica can be told how to assemble these solutions back into L?
  • I've tried a few variants on "Solve", putting the various entries of $L$ into curly braces.

Honestly, I'm pretty lost and my Google searches keep throwing up irrelevant pages. Any help would be hugely appreciated! Thanks in advance for any you can provide!

4 Replies

The system does not appear to have a solution.

xmat = {{a[1], 1}, {b[1], -a[1]}};
cmat = {{0, 1}, {b[1] - a[1]*a[1], 0}};

In[308]:= lmat = Array[l, Dimensions[xmat]];
Solve[xmat.lmat == lmat.cmat, Flatten[lmat]]

(* Out[309]= {{l[1, 1] -> 0, l[1, 2] -> 0, l[2, 1] -> 0, l[2, 2] -> 0}} *)

General remarl: avoid capital letters for variables. Many have built-in meanings.

Posted 4 months ago

Apologies, there was a sign error in my question. The cmat matrix should have been given by cmat={{0,1},{b[1]+a[1]*a[1],0}}.

Okay, so same idea. I changed lmat to be explicitly unimodular lower triangular.

lmat = {{1, 0}, {z, 1}};
Solve[xmat.lmat == lmat.cmat, Variables[lmat]]

(* Out[317]= {{z -> -a[1]}} *)
Posted 4 months ago

That works perfectly! Thank you so much for the help!

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