Kronecker product of matrices

APPENDIX A1 THE KRONECKER PRODUCT OF MATRICES AND THE vec(o) OPERATOR The Kronecker Product... 

This is unfortunate but this ordering is used in the standard mathematical conventions on the Kronecker products of matrices which will be used in the following theory. 

Other notation used diagB is the diagonal n x n matrix consisting of the diagonal elements of the square matrix B. The trace of B is denoted trB, and the determinant of B is denoted B. The Kronecker product of two matrices is denoted by symbol (g). Other notation will be introduced as needed. 

Wig51] Wigner EP 1951 On the matrices which reduce the Kronecker products of representations of simply-reducible groups (unpublished), cited in [App68]. 

Constructing an 50(4) matrix in terms of two SU(2) matrices parametrized by q and p is done as follows each of the SU(2) matrices corresponding to q and p, respectively, acts in a separate space of states of two particles with -spins [28,29]. Since the 50(4) group is a direct product of two 50(3) (or of SU(2) locally isomor-phous to 50(3)) groups the matrix representing an element of 50(4) is the direct (Kronecker) product of two SU(2) matrices. The space in which it acts is a direct product of two spaces spanned by the basis states +5), — 5) eac 1- configu-... 

The direct (Kronecker) product of the SU (2) matrices representing the q- and p-pararotations acts in this space with the notion that the q-dependent matrix eq. (3.48) acts on the states of the first particle and the p-dependent one on the states of the second particle in the product state. Then we form linear combinations of the above states, which correspond to specific values of the total spin and desired spatial symmetry. The combination which corresponds to the zero total spin of two particles transforms as a scalar i.e. (singlet) s-function. Those which correspond to the total spin equal to unity form the basis in the three-dimensional (triplet) space of />functions. The coordinate (x-, y-, and z-) functions are obtained as the following combinations of the states with the definite spin projections (the above product states) ... 

Hence, every bilinear component vec(arb7) can be written as a well-defined vector that is a function of ar and br only. This vector is the Kronecker product of br and ar. Define the Khatri-Rao product of two matrices, A and B, with the same number of columns as... 

Traaisformations of the basis now induce transformations of the operator and density matrices in terms of the Kronecker product of the transformation matrix with itself ... 

Wigner, E.P., 1965, On the Matrices which Reduce the Kronecker Product of Representations of S.R. [Simply Reducible] Groups, in Quantum Theory of Angular Momentum, eds L.C. Biedenham and H. van Dam (Academic Press, New York) p. 87. 

The outer or Kronecker product of two matrices (C = A X B) is defined by Cik ii = AijBki (double-index labelling of rows and columns). By writing a set of spin-orbitals in the form rl) = Xa, where < > is a row of orbitals and o = (spin factors are separately transformed. [Hint The matrix product of two outer products is... 

In the MO basis this is a product of Kronecker delta functions. In the case of the V intermediate needed in CCSD(F12) theory, the indices p and q are in the AO basis, and the overlap integrals are not the unity matrices anymore. Instead, they must be expressed as partially transformed overlap integrals... 

The Hadamard product of two matrices (A°B) produces another matrix where each element ij is the product of the elements ij of the original matrices. The condition is that A and B matrices must have the same dimensions. For instance, in the given examples of Figure 2 the Hadamard product could be done between A and B or even A and (where the superindex T denotes the transpose of the matrix), but not between A and C (Figure 3). The Kronecker product... 

FIGURE 3 Some examples of Hadamard, Kronecker and Khatri-Rao products of the matrices in Figure 2. 

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