Missing solutions of a matrix congruence

Question

There are some indications that the following matrix congruence might have a one-dimensional family of solutions. Also this discussion is relevant

a = DiagonalMatrix[{5, 4, 3}]
b = DiagonalMatrix[{3, 4, 5}]
X = Array[x, {3, 3}]
r = NSolve[
  Transpose[X].a.X == b && (Transpose[X].X)[[1, 1]] == 
    1 && (Transpose[X].X)[[2, 2]] == 1 && (Transpose[X].X)[[3, 3]] == 
    1, Flatten[X], Reals, WorkingPrecision -> 32]

This is clearly a solution

X={{0,0,1},{0,1,0},{1,0,0}}

I am curious if there are other real solutions. However, MA cannot find even the obvious one. Suggestion of experts how to obtain these (preferably the whole family) solutions are appreciated.

Answer 1

Rewrite your expression to remove the equalities (which are hard for real-valued equations) and represent it as a minimization problem. This gives an answer without trouble:

NMinimize[
 Norm[Flatten[{Thread[Transpose[X].a.X - b], (Transpose[X].X)[[1, 1]] - 
     1, (Transpose[X].X)[[2, 2]] - 1, (Transpose[X].X)[[3, 3]] - 1}]], Flatten[X]]