Commuting matrices

These conditions define a matrix algebra which requires at least four anticommuting, traceless (hence even-dimensioned) matrices. The smallest even dimension, n = 2, can only accommodate three anti-commuting matrices, the Pauli matrices ... 

Again IIW, M° and IM2n M2n, although IMX n + IM2n = M°Xn + M2 n. A product of non commuting matrices always corresponds to an order of decreasing indices, here m, from left to right. Matrices Nn 0 and Nn n are defined as unit matrices. Inversion of eqn. (102) yields... 

The same arguments can be extended in a stepwise fashion to any number of commuting matrices. 

As a result, the exact CC equations are quartic equations for the tjm, tjjm n, etc. amplitudes. Although it is a rather formidable task to evaluate all of the commutator matrix elements appearing in the above CC equations, it can be and has been done (the references given above to Purvis and Bartlett are especially relevant in this context). The result is to express each such matrix element, via the Slater-Condon rules, in terms of one- and two-electron integrals over the spin-orbitals used in determining d including those in itself and the virtual orbitals not in . 

We note that, even if we start here from the same truncated basis B = B, B2,. . . , Bm as in the EOM method, the results are not necessarily the same, since (2.16) is a single-commutator secular equation whereas (1.50) is a double-commutator secular equation. It should be observed, however, that the column vectors d obtained by solving (2.15) are optimal in the sense of the variation principle, whereas this is not necessarily true for the vectors obtained by solving (1.49). In the following analysis, we will discuss the connection between these two approaches in somewhat greater detail. Since the variation principle (2.10) would provide an optimal approximation, the essential question is whether the theoretical and computational resources available today would permit the proper evaluation of the single-commutator matrix elements defined by (2.13) for a real many-particle system this remains to be seen. 

It is therefore proven that only a constant matrix can commute with every matrix of a unitary irreducible representation. Example 2.14 deals with the fact that we need not have assumed that the commuting matrix be Hermitian. 

In the event that the representation by R,- is reducible, the ineihod of reduction is evident from E<. 4.9. The matrices R< are rcdu< ed by a similarity transformation with the matri.x A which brought the commuting matrix H into diagonal form, that is, the transformation R.-— R -for every i produces a reduced representation. Thus the R consist of nonzero diagonal elements and block diagonal entries such as in Eq. 4.2. 

Brandt,R.A. Physics on the Light Cone (Vol. 57) Dahmen.H.D. Local Saturation of Commutator Matrix Elements (Vol. 62)... 

It is more convenient to re-express this equation in Liouville space [8, 9 and 10], in which the density matrix becomes a vector, and the commutator with the Hamiltonian becomes the Liouville superoperator. In tliis fomuilation, the lines in the spectrum are some of the elements of the density matrix vector, and what happens to them is described by the superoperator matrix, equation (B2.4.25) becomes (B2.4.26). 

For a coupled spin system, the matrix of the Liouvillian must be calculated in the basis set for the spin system. Usually this is a simple product basis, often called product operators, since the vectors in Liouville space are spm operators. The matrix elements can be calculated in various ways. The Liouvillian is the conmuitator with the Hamiltonian, so matrix elements can be calculated from the commutation rules of spin operators. Alternatively, the angular momentum properties of Liouville space can be used. In either case, the chemical shift temis are easily calculated, but the coupling temis (since they are products of operators) are more complex. In section B2.4.2.7. the Liouville matrix for the single-quantum transitions for an AB spin system is presented. 

The normal rules of association and commutation apply to addition and subhaction of matrices just as they apply to the algebra of numbers. The zero matrix has zero as all its elements hence addition to or subtraction from A leaves A unchanged... 

It is helpful to remember that the element py is formed from the ith row of the first matrix and the jth column or the second matrix. The matrix product is not commutative. That is, AB BA in general. 

For a more complicated [B] matrix that has, say, n columns whereas [A] has m rows (remember [A] must have p columns and [B] must have p rows), the [C] matrix will have m rows and n columns. That is, the multiplication in Equations (A.21) and (A.22) is repeated as many times as there are columns in [B]. Note that, although the product [A][B] can be found as in Equation (A.21), the product [B][A] is not simultaneously defined unless [B] and [A] have the same number of rows and columns. Thus, [A] cannot be premultiplied by [B] if [A][B] is defined unless [B] and [A] are square. Moreover, even if both [A][B] and [B][A] are defined, there is no guarantee that [A][B] = [B][A]. That is, matrix multiplication is not necessarily commutative. 

This shows that, when we have found the correct electron density matrix and correctly calculated the Hartree-Fock Hamiltonian matrix from it, the two matrices will satisfy the condition given. (When two matrices A and B are such that AB = BA, we say that they commute.) This doesn t help us to actually find the electron density, but it gives us a condition for the minimum. 

Both in Eq. (8-149) and Eq. (8-147), we have written the function in the center of the integrand simply for ease of visual memory in fact both /(q) and F(q,q ) commute with all the B-operators and their positions are immaterial. The B-operators operate on vectors > in occupation number space, so that we can evaluate the matrix elements of F in occupation number representation, viz., Eq. (8-145), either from Eq. (8-147) or from Eq. (8-149). 

All other products of y-matrices can, by using the commutation rules, be reduced to one of these sixteen elements. The proof of their linear independence is based upon the fact that the trace of any of these matrices except for the unit matrix, I, is zero. If Tr is any one of these matrices, then rrr, generates again one of the T s, the unit matrix... 

Theorem B.—Any four-by-four matrix that commutes with a set of y is a multiple of the identity. 

The proof of this theorem follows from theorem A A four-by-four matrix that commutes with the y commuted with their products and hence with an arbitrary matrix. However, the only matrices that commute with every matrix are constant multiples of the identity. Theorem B is valid only in four dimensions, i.e., when N = 4. In other words the irreducible representations of (9-254) are fourdimensional. 

Summarizing, we have noted that the Heisenberg operators Q+(t) obey field free equations i.e., that their time derivatives are given by the commutator of the operator with Ha+(t) = Ho+(0) and that this operator H0+(t) is equal to H(t) = H(0). The eigenstates of H0+ are, therefore, just the eigenstates of H. We can, therefore, identify the states Tn>+ with the previously defined >ln and the operator [Pg.602]

The representation of these commutation rules is again fixed by the requirement that there exist no-particle states 0>out and 0>ln. The -matrix is defined as the unitary operator which relates the in and out fields ... 

If we restrict ourselves to the case of a hermitian U(ia), the vanishing of this commutator implies that the /S-matrix element between any two states characterized by two different eigenvalues of the (hermitian) operator U(ia) must vanish. Thus, for example, positronium in a triplet 8 state cannot decay into two photons. (Note that since U(it) anticommutes with P, the total momentum of the states under consideration must vanish.) Equation (11-294) when written in the form... 

Here it is taken into account that density matrix p, being a scalar, commutates with any rotation operator, and diq defined in Eq. (7.51) is used. After an analogous transformation, in master equation (7.51) there remains the Hamiltonian, which does not depend on e ... 

First consider the dipole operator (O = r). The matrix elements on rhs of eq. 17 are thus just the dipole transition moments, and the commutator becomes C = -ip. As the exact solution (complete basis set limit) to the RPA is under consideration, we may use eq. 10 to obtain... 

It is also of interest to study the "inverse" problem. If something is known about the symmetry properties of the density or the (first order) density matrix, what can be said about the symmetry properties of the corresponding wave functions In a one electron problem the effective Hamiltonian is constructed either from the density [in density functional theories] or from the full first order density matrix [in Hartree-Fock type theories]. If the density or density matrix is invariant under all the operations of a space CToup, the effective one electron Hamiltonian commutes with all those elements. Consequently the eigenfunctions of the Hamiltonian transform under these operations according to the irreducible representations of the space group. We have a scheme which is selfconsistent with respect to symmetty. 

The operation of matrix multiplication can be shown to be associative, meaning that X(YZ) = (XY)Z. But, it is not commutative, as in general we will have that XY YX. Matrix multiplication is distributive with respect to matrix addition, which implies that (X + Y)Z = XZ + YZ. When this expression is read from right to left, the process is called factoring-out [4]. 

If the product AB equals the product BA, then A and B commute. Any square matrix A commutes with the unit matrix of the same order... 

The parity matrix commutes with the first entropy matrix, gS = Sg, because there is no coupling between variables of opposite parity at equilibrium, (xiXj)0 = 0 if e,e - = — 1. If variables of the same parity are grouped together, the first entropy matrix is block diagonal. 

The odd expansion coefficients are block-adiagonal and hence c j I c I [g.3+ k3] = 0. This means that the coefficient of x on the right hand side is identically zero. (Later it will be shown that 0 and that could be nonzero.) Since the parity matrix commutes with the block-diagonal even coefficients, the reduction condition gives... 

Here we have used the symmetry and commuting properties of the matrices to obtain the final line. This shows that the correlation matrix goes like... 

It is important to note that the product of two square matrices, given by AB is not necessarily equal to BA. In other words, matrix multiplication is not commutative. However, the trace of the product does not depend on the order of multiplication. From Eq. (28) it is apparent that... 

As shown earlier, the operation is always in a class by itself, as it commutes with all other operations of the group. It is identified with Ti, the arbitrarily chosen first class of operation In a given representati.Qnt ie operation E corresponds to a. unit matrix whose order is equal to the dimension of the representation. Hence, the esultipg character, the sums of e diagonal elements, is also equal to the dimension of the representation, The dimension of each representation can thus be easily determined by inspection of the corresponding entry in the first column of characters in the table. 

Since it is necessary to represent the various quantities by vectors and matrices, the operations for the MND that correspond to operations using the univariate (simple) Normal distribution must be matrix operations. Discussion of matrix operations is beyond the scope of this column, but for now it suffices to note that the simple arithmetic operations of addition, subtraction, multiplication, and division all have their matrix counterparts. In addition, certain matrix operations exist which do not have counterparts in simple arithmetic. The beauty of the scheme is that many manipulations of data using matrix operations can be done using the same formalism as for simple arithmetic, since when they are expressed in matrix notation, they follow corresponding rules. However, there is one major exception to this the commutative rule, whereby for simple arithmetic ... 

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