Transforming a full $4times4$ symbolic matrix into a full $3times3$ matrix and an eigenvalue

I have a $4times4$ matrix

A=begin{pmatrix}
0.16 (cos (text{kx})+2) & 0.55 cos left(frac{text{kx}}{2}right)+(0., +0.76 i) sin left(frac{text{kx}}{2}right) & 0.55 cos left(frac{text{kx}}{2}right) & 0.55
0.55 cos left(frac{text{kx}}{2}right)-(0., +0.76 i) sin left(frac{text{kx}}{2}right) & 0.16 (cos (text{kx})+2) & 0.55 & 0.55 cos left(frac{text{kx}}{2}right)
0.55 cos left(frac{text{kx}}{2}right) & 0.55& 0.16 (cos (text{kx})+2) & 0.55 cos left(frac{text{kx}}{2}right)+(0., +0.76 i) sin left(frac{text{kx}}{2}right)
0.55 & 0.55 cos left(frac{text{kx}}{2}right) & 0.55 cos left(frac{text{kx}}{2}right)-(0., +0.76 i) sin left(frac{text{kx}}{2}right) & 0.16 (cos (text{kx})+2)
end{pmatrix}

A={{0.16*(2 + Cos[kx]), 0.55*Cos[kx/2] + (a + 0.76*I)*Sin[kx/2], 
      0.55*Cos[kx/2], 
  0.55 }, {0.55*Cos[kx/2] - (a + 0.76*I)*Sin[kx/2], 
      0.16*(2 + Cos[kx]), 0.55 , 0.55*Cos[kx/2]}, 
    {0.55*Cos[kx/2], 0.55, 0.16*(2 + Cos[kx]), 
      0.55*Cos[kx/2] + (a + 0.76*I)*Sin[kx/2]}, {0.55 , 
  0.55*Cos[kx/2], 
      0.55*Cos[kx/2] - (a + 0.76*I)*Sin[kx/2], 0.16*(2 + Cos[kx])}}

where $a$ and $k_x$ are real.

I would like to transform this matrix into a full $3 times 3$ matrix $F_{3times 3}(k_x,a)$ and a diagonal term $C(k_x,a)$, where $C$ is one of the eigenvalues of A, as

begin{equation}
A’=U^{-1}A U=begin{pmatrix}
F_{3 times 3} & 0_{3times 1}
0_{1 times 3} & C
end{pmatrix}.
end{equation}
where $U$ is the transforming matrix. When $U$ is a matrix with eigenvectors on its coloumns, the above transformation simply diagonalizes the full matrix and $F_{3 times 3}$ will be diagonal.
This particular $U$ is not what I am seeking, as I would like $F_{3times3}$ to be a full matrix with non-diagonal elements.
In principle, I can get this form of $A’$ using linear combinations of rows and columns. How can I perform such a transformation on A using eigenvectors of A?

  A = A /. kx -> k x

      DiagonalMatrix[
 Eigenvalues[A //. Cos[k x] -> ck /. Cos[k x/2] -> ck2] //. 
    Cos[k x] -> ck /. Cos[k x/2] -> ck2 // FullSimplify]