A Kronecker Product

The mass matrix M enters the Hamiltonian for convenience of expression and is an n X n matrix with on the diagonal elements and 2 on all of the off-diagonal elements the M notation for any matrix will mean a Kronecker product with the 3 X 3 identity matrix, M = M h. 

To maintain the parallel with 2-component theory, it is expedient to express the Dirac space as a Kronecker product of two Pauli spaces. R om... 

The natural extension of a dyad is a triad. The triad consists of a multiplication of three vectors a (/x 1), b(7 x 1) and c (K x l),andisvisualizedinFigure2.5.Themultiplication involved is sometimes referred to as a tensor product (A). In order to distinguish between Kronecker and tensor products, different symbols will be used. A Kronecker product is a tensor product, but not every tensor product is a Kronecker product [Burdick 1995], In... 

Effectively, vector r has 3 x n x 1 components since each r, in (47) is itself a three-dimensional vector. Technically speaking, in place of Ak in (46), one should write the Kronecker product A with being the 3 x 3 identity matrix. However, to simplify notations and avoid writing routinely this obvious Kronecker product, below in this section we will be using the following convention for matrix-vector multiplications involving such vectors ... 

The Kronecker product with the identity ensures rotational invariance (sphericalness) elliptical Gaussians could be obtained by using a full n x n A matrix. In the former formulation of the basis function, it is difficult to ensure the square integrability of the functions, but this becomes easy in the latter formulation. In this format, all that is required is that the matrix, A, be positive definite. This may be achieved by constructing the matrix from a Cholesky decomposition A), = Later in this work we will use the notation... 

A = A h to indicate the Kronecker product with the 3x3 identity matrix. 

T TW is called the direct product (or Kronecker product) of the representations r" and r The sign does not mean multiplication, it is simply a signal that the direct product of two representations has been formed in the manner given above. 

Appendix A. Composite indices. Some properties of direct (Kronecker) products. 284... 

APPENDIX A. COMPOSITE INDICES. SOME PROPERTIES OF DIRECT (KRONECKER) PRODUCTS... 

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) ... 

In the above equation the operation , called the Kronecker Product, is defined for A (an mxn matrix) and B (a p x q matrix) matrixes as partitioned mp x nq matrix whose ijth partition is ay times B ... 

Here r is a 3n x 1 vector of Cartesian coordinates for the n particles, Lk is an n x n lower triangular matrix of rank n and I3 is the 3x3 identity matrix, k would range from 1 to A where N is the number of basis functions. The Kronecker product with I3 is used to insure rotational invariance of the basis functions. Also, integrals involving the functions k are well defined only if the exponent matrix is positive definite symmetric this is assured by using the Cholesky factorization LkL k. The following simplifications will help keep the notation more compact ... 

The symmetric group S(n) is of fundamental importance in quantum chemistry as well in nuclear models and symplectic models of mesoscopic systems. One wishes to discuss the properties of the symmetric group for general n and concentrate on stable results that are essentially n—independent. Here the reduced notation(6)-(9) proves to be very useful. The tensor ir-reps A of S(n) are labelled by ordered partitions(A) of integers where A I- n. In reduced notation the label Ai, A2,. .., Ap for S(n) is replaced by (A2,...,AP). Kronecker products can then be fully developed in a n-independent manner and readily programmed. Thus one finds, for example, the terms arising in the reduced Kronecker product (21) (22) are... 

Looking at the above list one is immediately struck by the observation that the list is self-associated. That is every partition (A) in the list either has a conjugate partner (A) where the rows and columns of the Young frame of the partition (A) have been interchanged or the partition (A) is self-conjugate. Some Kronecker products are self-associated while others are not. Is there... 

A number of problems arise in studying the properties of infinite dimensional irreps of Sp(2d, R) in order to make practical applications. These include evaluating Kronecker products and resolving symmetrised powers of the basic irreps ( (0)) and ( (1)). The Kronecker products have been discussed elsewhere(ll)-(13). The resolution of the symmetrised powers of the basic irreps is a particularly difficult problem and until now no general results have been known. The symmetrised squares of the basic irreps of Sp(2d, R) have recently been studied in some detail for various values of d and up to terms of weight 20. This led me to guess that in general... 

In order to concisely describe multi-way models, the usual matrix product is not sufficient. Three other types of matrix products are introduced the Kronecker ( ), Hadamard ( ) and Khatri-Rao (O) product [McDonald 1980, Rao Mitra 1971, Schott 1997], The Kronecker product allows a very efficient way to write Tucker models (see Chapter 4). Likewise, the Khatri-Rao product provides means for an efficient way to write a PARAFAC model (see Chapter 4). The Hadamard product can, for instance, be used to formalize weighted regression (see Chapter 6). 

In the example used earlier for the Kronecker product, the Hadamard product of A and B becomes ... 

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... 

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

A similar transformation is induced in the electron-repulsion integrals which may be written in Kronecker-product form or, more commonly, simply written out in full ... 

Once the electric or magnetic dipole nature of the transitions is known, selection rules are determined from group theory. The components of the electric and magnetic field vectors transform according to definite representations of the point group (often different ones, as E is a polar vector and B is an axial vector). A particular transition is allowed (forbidden) if the irreducible representation of the final state is (is not) contained in the Kronecker product of the initial state representation and the representation of the appropriate component of E or B (Tinkham, 1%4). 

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