# Why can't NSolve solve my system of equations?

Here is the continuation of my previous post, but now, we have special case of the problem and all values are numerically presented. Only, the system of unknowns T I need to solve with conditions given in Solve. Is there any solution to solve this system numerically or analytically, it will be nice if I can include Parallelize command, but I don't know how

unknown

T = {{T11, T12, T13, T14}, {T21, T22, T23, T24}, {T31, T32, T33, T34},
{T41, T42, T43, T44}};

known

JJ = {{0, 0, 1, 0}, {0, 0, 0, 1}, {-1, 0, 0, 0}, {0, -1, 0, 0}};

H = {{1112000 π^2 + (400 π^4)/21,
11120000 π - (12800 π^3)/21, 0, 0},
{11120000 π - (12800 π^3)/21,
111200000 + (544000 π^2)/21, 0, 0},
{0, 0, 272/( 525 (1088/21 + (16 π^2)/328125)),
(32 π)/(2625 (1088/21 + (16 π^2)/328125))},
{0, 0, (32 π)/(2625 (1088/21 + (16 π^2)/328125)),
(100 + π^2/2625)/(1088/21 + (16 π^2)/328125)}};

v1 = 115670807 /5868910;
v2 = 3711603512 /252957;

V = {{v1, 0, 0, 0}, {0, v2, 0, 0}, {0, 0, v1, 0}, {0, 0, 0, v2}};

So now I need to solve the simple system *(16 unknowns), and it doesn't work. Is there any method to determine T?

NSolve[{Transpose[T].H.T - V == 0, Transpose[T].JJ.T - JJ == 0},
Flatten@T]
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Not every equation has a solution. Are you sure this set has one? –  Sjoerd C. de Vries Jul 16 at 15:00
it should be, this is a part of standard contact transformation method, but I couldn't find exact tutorial or example to test it. Is there any already prepared method in package mathematica to test this system –  Pipe Jul 16 at 15:04
Do I smell P vs NP :D :D Umm I am trying right now to compute Reduce[{Transpose[T].H.T - V == 0, Transpose[T].JJ.T - JJ == 0}, {T11, T12, T13, T14, T21, T22, T23, T24, T31, T32, T33, T34, T41, T42, T43, T44}] –  Sektor Jul 16 at 15:12
I'm fairly confident that if Solve yields the empty solution set without any warnings, there is in fact no general solution. –  Sjoerd C. de Vries Jul 16 at 15:15
I get 32 eqns, 16 variables. So it is overdetermined. Are you certain the system is consistent? –  Daniel Lichtblau Jul 19 at 20:36