Change of basis matrix between symmetric matrices

Question

I have two 5x5 matrices:

A={{430.984, -197.898, 106.409, -212.614, -61.7758}, {-197.898, 1144.67, -397.431,
    26.8918, 122.652}, {106.409, -397.431,
      492.869, -213.863, -122.323}, {-212.614, 26.8918, -213.863,
      960.716, -292.851}, {-61.7758, 122.652, -122.323, -292.851,
      492.797}}
B={{428.257, -110.756, -147.675, -110.756, -44.3025}, {-110.756,
      904.509, -369.187, -276.891, -110.756}, {-147.675, -369.187,
      1082.95, -369.187, -147.675}, {-110.756, -276.891, -369.187,
      904.509, -110.756}, {-44.3025, -110.756, -147.675, -110.756,
      428.257}}

They are symmetric.

What is the fastest way to find the change of basis matrix $P$ that solves for $P^TAP=B$?

mX = (Array[X, {5, 5}] );

NSolve[FullSimplify[Transpose[mX].A.mX] == B, Flatten[mX]]

does not end.

Answer 1

a = Eigensystem[A];
b = Eigensystem[B];

P = Transpose[a[[2]]] . DiagonalMatrix[Sqrt[b[[1]]/a[[1]]]] . b[[2]];

Transpose[P] . A . P - B // Norm
(*    4.31222*10^-12    *)