Interaction of quantum system S a described by the generalised ρ×ρ eigenvalue equation A|Θ s 〉=E s S a |Θ s 〉 (s=1,...,ρ) with quantum system S b described by the generalised n×n eigenvalue equation B|Φ i 〉=λ i S b |Φ i 〉 (i=1,...,n) is considered. With the system S a is associated ρ-dimensional space X ρ a and with the system S b is associated an n-dimensional space X n b that is orthogonal to X ρ a . Combined system S is described by the generalised (ρ+n)×(ρ+n) eigenvalue equation [A+B+V]|Ψ k 〉=e k [S a +S b +P]|Ψ k 〉 (k=1,...,n+ρ) where operators V and P represent interaction between those two systems. All operators are Hermitian, while operators S a ,S b and S=S a +S b +P are, in addition, positive definite. It is shown that each eigenvalue e k ∉λ i of the combined system is the eigenvalue of the ρ×ρ eigenvalue equation $$[\Omega (\varepsilon _k ) + A]|\Psi _k^a \rangle = \varepsilon _k S^a |\Psi _k^a \rangle $$ . Operator $$\Omega (\varepsilon )$$ in this equation is expressed in terms of the eigenvalues λ i of the system S b and in terms of matrix elements 〈χ s |V|Φ i 〉 and 〈χ s |P|Φ i 〉 where vectors |χ s 〉 form a base in X ρ a . Eigenstate |Ψ k a 〉 of this equation is the projection of the eigenstate |Ψ k 〉 of the combined system on the space X ρ a . Projection |Ψ k b 〉 of |Ψ k 〉 on the space X n b is given by |Ψ k b 〉=(e k S b −B)−1(V−e k P})|Ψ k a 〉 where (e k S b −B)−1 is inverse of (e k S b −B) in X n b . Hence, if the solution to the system S b is known, one can obtain all eigenvalues e k ∉λ i } and all the corresponding eigenstates |Ψ k 〉 of the combined system as a solution of the above ρ×ρ eigenvalue equation that refers to the system S a alone. Slightly more complicated expressions are obtained for the eigenvalues e k ∈λ i } and the corresponding eigenstates, provided such eigenvalues and eigenstates exist.