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At present, I want to generate some random matrices whose nth power is zero matrix.

mat0[n_] := 
 Module[{k}, 
  While[MatrixPower[k = RandomInteger[{1, 5}, {n, n}], 3] =!= 
    ConstantArray[0, {n, n}]]; k]
MatrixPower[#, 3] & /@ Table[mat0[3], 3]

A = Array[x, {3, 3}];
FindInstance[MatrixPower[A, 3] == ConstantArray[0, {3, 3}], 
 Flatten[A], Reals, 3]

The above code is used to get the matrix whose third power is zero matrix. But the above code has been running, what can I do to quickly get the random matrix that meets the requirements?

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1 Answer 1

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I make no claims whatsoever on the distribution followed by the matrices generated by the following routine:

randomNilpotentMatrix[n_Integer?Positive, p_Integer?Positive] := 
      With[{uf = First[HermiteDecomposition[RandomInteger[{-9, 9}, {n, n}]]]}, 
           LinearSolve[uf, SparseArray[{Band[{1, 2}] -> ConstantArray[1, Min[p, n] - 1]},
                                       {n, n}]].uf]

For instance,

BlockRandom[SeedRandom[42]; randomNilpotentMatrix[3, 3]]
   {{73, 109, -116}, {283, 419, -444}, {313, 464, -492}}

MatrixPower[%, 3]
   {{0, 0, 0}, {0, 0, 0}, {0, 0, 0}}
   {{0, 0, 0}, {0, 0, 0}, {0, 0, 0}}
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  • $\begingroup$ Thank you very much. How to modify your code to generate a matrix whose power of 4 is not zero matrix but whose power is 5 is zero matrix? $\endgroup$
    – A little mouse on the pampas
    Commented Aug 14, 2020 at 1:49
  • $\begingroup$ Try randomNilpotentMatrix[5, 5]. $\endgroup$
    – J. M.'s missing motivation ♦
    Commented Aug 14, 2020 at 1:51
  • $\begingroup$ Is it possible to generate a 3 * 3 matrix whose 4th power is not zero matrix but its 5th power is zero matrix? $\endgroup$
    – A little mouse on the pampas
    Commented Aug 14, 2020 at 1:54
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    $\begingroup$ It's a good linear algebraic exercise for you to prove that your request is unreasonable. $\endgroup$
    – J. M.'s missing motivation ♦
    Commented Aug 14, 2020 at 1:56
  • $\begingroup$ Thank you very much. According to the Hamilton-Cayley theorem, we can know that the nilpotent exponent is not greater than the order of the matrix. $\endgroup$
    – A little mouse on the pampas
    Commented Aug 14, 2020 at 2:28

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