equation solving - Finding an Integer, Unimodular Matrix that connects two given matrices

I have two symmetric, integer matrices, $K$ and $K_2$ which have the same determinant and the same signature (number of positive - number of negative eigenvalues). I want to find an integer valued matrix, $X$ , such that

$X^T K X= K_2$ and determinant of

$X = \pm 1$ .

For example,

K = {{0, 2, 1, 1}, {2, 0, 0, 0}, {1, 0, 1, 0}, {1, 0, 0, -1}};
K2 = {{0, 2, 0, 0}, {2, 0, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, -1}};

Is there a way to approach this besides using FindInstance (which doesn't seem to work in finite time for even a $4\times4$ matrix)? The system is under constrained since the matrices are symmetric, but I'm just looking for a way to find even one such $X$ for a given pair, $K$ and $K_2$ , which could be as large as $8\times8$ .

Answer

Using Smith Normal Form you can get two integer matrices with determinant 1 that would satisfy the equation

$X1.K.X2 = K_2$

May be this is a good start you can use...

resK = SmithDecomposition[K];
MatrixForm /@ resK

resK2 = SmithDecomposition[K2];

MatrixForm /@ resK2

resK[[2]] == resK2[[2]]
(* True *)

K2 == 
 Inverse[resK2[[1]]].resK[[1]].K.resK[[3]].Inverse[resK2[[3]]]
(* True *)

X1 = Inverse[resK2[[1]]].resK[[1]];


X2 = resK[[3]].Inverse[resK2[[3]]];

X1.K.X2 == K2
(* True *)

Det /@ {X1, X2}
(* {1, 1} *)

MatrixForm /@ {X1, X2}

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