Direct product of two matrices

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

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, = ... 

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) ... 

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. 

While the matrix multiplication defined by Eq. (28) is the more usual one in matrix algebra, there is another way of taking the product of two matrices. It is known as the direct product and is written here as A <8> 1 . If A is a square matrix of order n and B is a square matrix of order m, then A<8>B is a square matrix of order tun. Its elements consist of all possible pairs of elements, one each from A and B, viz. 

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

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 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. 

All direct methods for solution of the system of linear equations (3) are based on the decomposition of matrix N into the product of two matrices. 

It follows from the representation theory of groups that the direct product of two irreducible representations can be decomposed into direct sum of irreducible representations of the same group. In case of the SO(3) group, the direct product of two Wigner-matrices can be decomposed into a direct sum of Wigner-matrices in the form... 

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

There is also a relation between polar unit vectors, boost generators, and electric fields. An electric field is a polar vector, and unlike the magnetic field, cannot be put into matrix form as in Eq. (724). The cross-product of two polar unit vectors is however an axial vector k, which, in the circular basis, is e<3>. In spacetime, the axial vector k becomes a 4 x 4 matrix related directly to the infinitesimal rotation generator /3) of the Poincare group. A rotation generator is therefore the result of a classical commutation of two matrices that play the role of polar vectors. These matrices are boost generators. In spacetime, it is therefore... 

All resolution methods mathematically decompose a global instrumental response of mixtures into the contributions linked to each of the pure components in the system [1-10]. This global response is organized into a matrix D containing raw measurements about all of the components present in the data set. Resolution methods allow for the decomposition of the initial mixture data matrix D into the product of two data matrices C and ST, each of them containing the pure response profiles of the n mixture or process components associated with the row and the column directions of the initial data matrix, respectively (see Figure 11.2). In matrix notation, the expression for all resolution methods is ... 

In view of the cominutability of A and B elements we may write the matrix product of two direct product matrices as follows ... 

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... 

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 ... 

Rather than directly calculating the product A b, we use a process called LU factorisation, to factor A as the product of two other matrices a lower and an upper triangular matrix, so that A = LU ... 

The two-component spinor functions cpfi can be chosen as spin orbitals which are the direct product of real scalar functions with spin functions, A (x) a,jS. Such a choice reduces the cost of basis orthogonalization since only real matrices are involved and different spins are decoupled. 

It is clear then that one needs to describe all relevant operators (related to the spin components) using their matrix representations in the four-fold vector space generated by the state vectors mj, ms). These matrices can be constructed by evaluating each matrix element or they can be built by the direct tensorial product of the corresponding 2x2 matrices that describe the dynamics of the separate spin 1 /2 systems [5,12], The direct tensorial product of two nxn matrices A and B lead to a x matrix C = A B whose elements are... 

Exactly the same formulation is constructed for the other P—O bond (B) then the two expressions are combined to give the time dependence of the joint conditional probabilities. This combination results in a lengthy nine-term equation, the terms of which are expressed as direct products of the W matrices, equilibrium populations, and exponential decay factors but are no more complex in principal than their analogs in the single two-state treatment. The first two terms are... 

Before showing further applications of direct-product representations to quantum mechanics, we quote without proof a theorem we will need. Let rij a and rkip be the elements of the matrices corresponding to the symmetry operation R in the two different nonequivalent irreducible representations Ta and T it can be shown that... 

---