Find an unknown matrix from an equation

I need to find an unknown matrix, H, from this equation:

K2 = Transpose[H].K1.H.

K1 and K2 are 2×2 matrices, and H is the unknown matrix, also 2×2. I know the first element of H, H(1,1)=1.

Can I simply use of some Mathematica functions? Because I tried with Solve:

Solve[K2==Transpose[H].K1.H] 

Doesn’t work.

Please help me.


Specifically I have this matrices:

 K1 = {{10385.46, -306.12}, {-306.12, 225.8}};  K2 = {{10049.794, 378.59}, {378.59, 1807.16}};  H = {{1, h12}, {h21, h22}}; 

For which I have this warning:

Solve was unable to solve the system with inexact coefficients. The answer was obtained by solving a corresponding exact system and numericizing the result