Direct product, of matrices

The arrangement of die elements in the direct-product matrix follows certain conventions. They are illustrated in the following chapter, where the direct product of matrices is employed in the theory of groups. 

The operation of direct product of matrices is both associative and also distributive with respect to matrix addition, and hence finally... 

This is the desired result The character of any symmetry operation in the direct-product representation TC is the product of its characters in the representations TF and Tc. (The direct product of matrices is not, in general, commutative however, A<8>B and B A have equal traces, and thus the corresponding direct-product representations are equivalent to each other.)... 

From Eq. 2.15 the trace of the direct product of matrice.s is equal to the product of the traces. Thus, the character of the direct product repre.sentation is obtained by multiplying the characters of the two repre.sentations. From the Dah character table we have the following ... 

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

The direct product of two matrices is best explained in terms of an example. 

The concept can once again be extended to the direct product of more than two matrices. 

In order to evaluate the spectral density of Eq. (35) or (38), one needs a complete basis set spanning the lattice operator space. This basis set can be obtained by taking direct products of Wigner rotation matrices,... 

The direct product of two matrices is quite different from the ordinary matrix product. First, let us consider how the indices of the various matrix elements are related, By comparing eqns (8-3.2) and (8-3.4) we have f, = ... 

The symmetric direct product of a one-dimensional representation with itself is clearly the same as the ordinary direct product of the representation with itself if TF is one-dimensional, then its matrices are the same as its characters, so that... 

For the direct product of two one-dimensional representations, the direct-product matrices are the same as the characters.)... 

Such a matrix adopts a block-diagonal form each block is a direct product of the rotational matrices for spatial and spin variables. 

Using the concepts derived above the density matrices of the products, at the moment of their formation, can be presented as the direct products of the density matrices of the molecular fragments involved (independently of the choice of basis function sets in the spaces of the reagents), e.g. ... 

In the direct product of the density matrices in equation (72), one may therefore neglect the term 6, d, which is of the order of (hcojkT)2 without any appreciable loss in accuracy ... 

Let us denote by the symbol Am n a matrix with m rows and n columns. The symbols A1,, A" , and A1,n then denote a scalar, a column vector, and a row vector, respectively. The direct product of two matrices is defined by equation (A4) (see, for example, reference 46) ... 

Direct multiplication of matrices, which are subscripted with composite indices, is carried out in the same manner as that of doubly-subscripted matrices e.g. the direct product in equation (93a) can be expressed as ... 

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 product of two matrices Q = P X R is a matrix whose dimensionality is the product of the dimensionalities of the two matrices. The components of Q consist of all products of the components of the separate matrices, P >Rm , with a convention as to ordering of the resultant components in Q. A specific example is given in Appendix D. 

The basic 2X2 spin matrices for one spin can be expanded to 4 X 4 matrices for an I— S spin system by taking the direct product of each with the 2X2 unit matrix, keeping the order I before S. For example,... 

Here vA is the Liouville space vector corresponding to v> direct product of the system aa state and the bath aa state. For subsequent manipulations we further introduce the system and the bath density matrices corresponding to thermal equilibrium within the electronically excited state, that is,... 

The computational procedure involves obtaining the matrix representation of the symmetry operators of the (2n + 2) site chain in the direct product basis. The matrix representation of both J and P for the new sites in the Fock space is known from their definitions. Similarly, the matrix representations of the operators J and P for the left (right) part of the system at the first iteration are also known in the basis of the corresponding Fock space states. These are then transformed to the density matrix eigenvectors bcisis. The matrix representation of the symmetry operators of the full system in the direct product space are obtained as the direct product of the corresponding matrices ... 

The direct product of two diagonal matrices is a diagonal matrix. This follows by iii.spcction of Kcp 2..51 since for a diagonal matri.x the only nonzero matrix elements are a,-,- and h k- The direct product of two unit matrices is a unit matrix, as may be verified by inspection of Kc). 2.50. 

The result in Eq. 5.75 follows from matrix theory—the matrix product of two direct products is the direct product of the two matrix products. Thus, according to Eq. 5.75 the direct product matrices A X B form a representation of dimension of the direct product group. The representations in Eq. 5.75 are irreducible. One may be concerned as to whether the direct products will yield all the irreducible representations of the direct product group. For the groups A and B we know that... 

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