Matrix representation of the noninteracting eigenvalue problem

It is of some interest to examine the matrix representation of the variational conditions for the fragment states and the compound state. Thus, the variational conditions for the compound system (4.3.29) may be written in the form 

For a compact representation of the eigenvalue equations for noninteracting systems, we introduce the direct products of matrices and vectors 

When A and B are vectors, the second index in (4.3.43) is absent or equal to unity (A,i and Bn). The Hamiltonian matrix with elements (4.3.42) may now be written in the form 

According to the preceding discussion, the solutions to the equations (4.3.41) with a separable Hamiltonian matrix (4.3.44) may be obtained in the form of the direct products 

To verify the direct-product form of the solutions of (4.3.41), we first note the identity 

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