Change of representations

Assume {|u_i>} and {|t_i>} are orthonormal bases for a Hilbert space.

Let {|u_i>} = old basis,
{|t_i>} = new basis,
and let U|u_i> = |t_i> be the unitary transformation from the old to new basis.
Then <u_i|U|u_j> = <u_i|t_j> = U_ij = matrix elements of U.

Assume |ψ> = ∑_ja_j|u_j> = ∑_jd_j|t_j> is an arbitary vector in the vector space.

What is the relationship between the components of this vector in the two bases?
<t_i|ψ> = <t_i|∑_jd_j|t_j> = d_i = ∑_ja_j<t_i|u_j> = ∑_jU^†_ija_j.

d_i =  ∑_jU^†_ija_j = components in new basis in terms of the components in the old basis.
a_i =  ∑_jU_ijd_j = components in old basis in terms of the components in the new basis.

What about operators?

Let |Φ> = Ω|ψ>, |ψ> = ∑_ja_j|u_j>,  |Φ> = ∑_ja_j'|u_j>.
∑_ja_j'|u_j> = ∑_ja_jΩ|u_j>.
a_i' = ∑_j<u_i|Ω|u_j>a_j = ∑_jΩ_ija_j,
In a given representation, the matrix of Ω is defined by the action of Ω on the basis vectors.

What is the relationship between the matrix elements of an operator Ω in the two bases {|u_i>} and {|t_i>}?
<t_i|Ω|t_j> = <t_i|UU^†ΩUU^†|t_j> = <u_i|U^†ΩU|u_j>, or Ω(new basis)_ij = (U^†ΩU)(old basis)_ij.
<u_i|Ω|u_j> = <u_i|U^†UΩU^†U|u_j> = <t_i|UΩU^†|t_j>, or Ω(old basis)_ij = (UΩU^†)(new basis)_ij.

How do we find the elements of the matrix of U that changes the representation from a different basis {|u_i>} to the eigenbasis of the Hermitian operator Ω?

Assume that Ω(old basis)_ij is given in the the {|u_i>} basis, and it is not diagonal.
Assume Ω(new basis)_ij is diagonal, the {|t_i>} are eigenvectors of Ω.

To find the matrix of U we find the eigenvalues and expansion coefficient of the eigenvectors of Ω in the old basis.  This is the typical eigenvalue problem.

Let a_i^j be the ith expansion coefficient of the jth eigenvector in the {|u_i>} bases.
In the {|t_i>} the ith expansion coefficient of the jth eigenvector is just δ_ij.
Then a_i =  ∑_kU_ikd_k  -->  a_i^j  = ∑_kU_ikδ_kj = U_ij, which shows that the columns of the matrix of U are the expansion coefficients of the different eigenvectors in the {|u_i>} basis, i.e. U_ij = a_i^j.

The matrix of Ω in the {|t_i>} basis is diagonal.  It has the eigenvalues on its diagonal.

Note: What you call U and what you call U^† depends on which basis you consider the old basis and which basis the new basis.
If the old basis is the eigenbasis and the new basis is a different basis, then U and U^† are reversed.