Implementing Brockett's Flow in Mathematica

Question

Motivated by this post and this paper is an attempt to implement this simple relation in Mathematica to diagonalize a $(2,2)$ matrix. However, the result is not diagonal after iterating discretized version 100k times, can anyone see the correct way to implement this?

brockett = 
  Compile[{{X0, _Real, 2}, {A, _Real, 
     2}, {lr, _Real}, {iters, _Integer}}, 
   Module[{i, comm, X}, X = X0;
    For[i = 1, i <= iters, i += 1, comm = X . A - A . X;
     X += lr  (X . comm - comm . X);];
    X]];
X0 = {{3, 1}, {1, 2}} // N;
A = {{5, 2}, {2, 7}} // N;
{"A", A // MatrixForm}
{"X0", X0 // MatrixForm}
{"Xt", brockett[X0, A, .0001, 100000] // MatrixForm}

Answer 1

Looks like A needs to be diagonal.

Also, couldn't figure out why I couldn't reuse definition of commutator inside the compiled function until reading this post, I guess I need to compile expressions first in order to inline them.

ClearAll["Global`*"];
SeedRandom[1];
(* commutator )
comm = Compile[{{X, _Real, 2}, {Y, _Real, 2}}, X . Y - Y . X];
( symmetrizer *)
symm[X0_] := (X0 + X0[Transpose])/2;
d = 10;
X0 = symm[RandomVariate[NormalDistribution[], {d, d}]/Sqrt[d]];
A = DiagonalMatrix[RandomVariate[NormalDistribution[], {d}]]/Sqrt[d];
brockett = 
  Compile[{{X0, _Real, 2}, {A, _Real, 
     2}, {lr, _Real}, {iters, _Integer}},
   Module[{i, X},
    X = X0;
    For[i = 1, i <= iters, i += 1,
     X += lr   comm[X, comm[X, A]];
     ];
    X
    ],
   CompilationOptions -> {"InlineExternalDefinitions" -> True}
   ];
relError[a_, b_] := Norm[a - b, 1]/Min[Norm[a, 1], Norm[b, 1]];
Xt = brockett[X0, A, .2, 50000];
Print["eig approx error: ", 
  relError[Sort@Diagonal[Xt], Sort@Eigenvalues[X0]]/d];
Print["Xt offdiagonal: ", Total[Abs@UpperTriangularize[Xt, 1], 2]];
ListLinePlot[{Sort@Eigenvalues@X0, Sort@Diagonal@Xt}, 
 PlotLegends -> {"true eigs", "Xt diagonal"}]
ArrayPlot@Xt