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The code below is given in Eric Weisstein's Math World article: Matrix Minimal Polynomial.

MatrixMinimalPolynomial[a_List?MatrixQ,x_]:=Module[
    {
      i,
      n=1,
      qu={},
      mnm={Flatten[IdentityMatrix[Length[a]]]}
    },
    While[Length[qu]==0,
      AppendTo[mnm,Flatten[MatrixPower[a,n]]];
      qu=NullSpace[Transpose[mnm]];
      n++
    ];
    First[qu].Table[x^i,{i,0,n-1}]
  ]

Is it possible to adapt the code to give the minimal polynomial of a matrix over a finite field of prime order $p$. Perhaps by adding Modulus -> p in the argument for NullSpace.

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    $\begingroup$ Yes, it seems to work fine with adding a Modulus argument to NullSpace. What is your question? $\endgroup$
    – Roman
    Commented Nov 19, 2022 at 16:07
  • $\begingroup$ Hi Roman. Thanks. The reason I am asking is that I am not clear on what the code is actually doing and consequently I am not sure if just adding a Modulus argument will work. $\endgroup$
    – geoffrey
    Commented Nov 19, 2022 at 16:34

1 Answer 1

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This can be done in such a way:

ResourceFunction["MatrixMinimalPolynomial"][{{22,-16},{25,-18}}, x,Extension->alg,Modulus->5]

1 + 4*x + 4*x^2

Of course, the above matrix is treated by mod 5. The theory is stated here.

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    $\begingroup$ This is fine (as in +1). Just wanted to add that the Extension option can be omitted if there is no algebraic extension under consideration. $\endgroup$
    – Daniel Lichtblau
    Commented Nov 19, 2022 at 17:32
  • $\begingroup$ @DanielLichtblau: No, in 13.1 on Windows 10 ResourceFunction["MatrixMinimalPolynomial"][{{22, -16}, {25, -18}}, x, Modulus -> 5] produces Last[{}] . {1} and the error "Last::nolast: {} has zero length and no last element". The documentation on this option is unclear. $\endgroup$
    – user64494
    Commented Nov 19, 2022 at 18:11
  • $\begingroup$ Hi user64494. Thanks. I accepted your answer already but I think there is a problem. For example: Let the matrix over $\mathbb{F}_2$ be {{0,1,0,0},{1,1,0,0},{0,0,0,1},{0,0,1,1}}. The minimal polynomial should be $x^2 + x + 1$. But using your code I get $x^3 + 1. $\endgroup$
    – geoffrey
    Commented Nov 19, 2022 at 20:03
  • $\begingroup$ Thanks both of you for pointing out those bugs. I'll look into them. $\endgroup$
    – Daniel Lichtblau
    Commented Nov 20, 2022 at 21:41
  • $\begingroup$ I updated` ResourceFunction["MatrixMinimalPolynomial"]` to fix the two issues noted here. I hope there are no further issues (but of course will look into any that get noted). $\endgroup$
    – Daniel Lichtblau
    Commented Nov 23, 2022 at 15:11

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