Matrices Hermitian form

If A is a square matrix and AT is a column matrix, the product AX is a so a column. Therefore, the product XAX is a number. This matrix expression, which is known as a quadratic form, arises often in both classical and quantum mechanics (Section 7.13). In the particular case in which A is Hermitian, the product XxAX is called a Hermitian form, where the elements of X may now be complex. 

The other theorem states that the matrix X formed by using the eigenvectors of a Hermitian matrix as its columns is unitary (for the definition of a unitary matrix, see Appendix A.4-l(ff)). The proof of these two theorems is given in Appendix A.4-3. 

Equation (149) is valid provided that the matrix T and that within the brackets are non-singular. It has been shown that this is always true in the case considered. (47) Since Newmark and Sederholm have used the exchange superoperator in its non-Hermitian form, such as that given in equation (117), their method is rather time-consuming. Probably, this is the reason why the method has been largely forgotten. 

The apparent non-Hermitian relationship (i.e., T// 7 Tf, even though Hu = //n and Sif = Sfi), is an artifact of the Condon factorization of the full vibronic matrix element, and, in a complete vibronic model, the vibrational manifold would compensate the mismatch in //, and //// [6, 60], In using the Condon-factorized form (Eqs. 72 and 73), one may simply bypass the problem by using a mean value (//) for Ha and Hp, thereby reverting to the Hermitian form of Eq. 69. 

This can also be stated in words that an operator A usually operates to the right on a function but it can also operate to the left on the complex adjoint i j = Hermitian operator. This also means that in the matrix mechanics form of quantum mechanics a given matrix-element of a Hermitian matrix is related to another element on the other side of the (upper left to lower right) diagonal of the matrix by the relationship Amn = A. The following theorem shows why this definition is useful. 

The set of eigenveetors of any Hermitian matrix form a eomplete set over the spaee they span in the sense that the sum of the projeetion matriees eonstrueted from these eigenveetors gives an exaet representation of the identity matrix. 

A Hessenberg form H (the same form but not the same matrix) can also be obtained by a sequence of orthogonal transformations, either by plane rotations (the method of Givens), each rotation annihilating an individual element, or by using unitary hermitians, I — 2wiwf, wfwt = 1 (the method of Householder), each of which annihilates ill possible elements in a column. Thus, at the first step, if A = A, and... 

For general matrices the reduction by triangular matrices requires less computation and is probably to be preferred. But if A is hermi-tian, observe that the use of the unitary reduction produces a matrix H that is again hermitian, hence, that is tridiagonal in form, having zeros everywhere except along, just above, and just below the main diagonal. 

The equation defines V(u), of course, since H is understood to be H(u). The matrix V is unitary. To see this, multiply (7-53) by l/t on the left then multiply the associate (or adjunct) of (7-53), —MVt — Wffi, by U on the right and note that H is hermitian W = H. Subtracting one of the resulting forms from the other gives the further result... 

Our theorem permits the following inference. The statistical matrix of every pure case in quantum mechanics is equivalent to an elementary matrix and can be transformed into it by a similarity transformation. Because p is hermitian, the transforming matrix is unitary. A mixture can, therefore, always be written in the diagonal form Eq. (7-92). 

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... 

To determine the form of M it is assumed to have a general Hermitian 2x2 matrix structure,... 

Note that, in contrast to other forms of intermolecular perturbation theory to be considered below, the NBO-based decomposition (5.8) is based on a full matrix representation of the supermolecule Hamiltonian H. All terms in (5.8) are therefore fully consistent with the Pauli principle, and both /7units(0, and Vunits(mt) are properly Hermitian (and thus, physically interpretable) at all separations. 

This leads to an exchange-correlation potential in the form of a 2 x 2 Hermitian matrix in spin space... 

Following Ref. [5] the T1 condition is obtained by considering an operator A = Y ij gij,kaiajak, where the gij k are arbitrary real or complex coefhcients totally antisymmetric in the three indices. (We view g as a vector of dimension (0, where r is the size of the one-electron basis.) The contractions (t / A+A t /) and (t / AA+ t /) both involve the 3-RDM, but with opposite sign, and so the nonnegativity of (tk 4 4 -f AA I ) for all three-index functions g provides a representability condition involving only the 1-RDM and 2-RDM. In exphcit form the condition is of semidefinite form, 0 T, where the Hermitian matrix T is... 

The preceding is a rather comprehensive—but not exhaustive— review of N-representability constraints for diagonal elements of reduced density matrices. The most general and most powerful V-representability conditions seem to take the form of linear inequalities, wherein one states that the expectation value of some positive semidefinite linear Hermitian operator is greater than or equal to zero, Tr [PnTn] > 0. If Pn depends only on 2-body operators, then it can be reduced into a g-electron reduced operator, Pq, and Tr[Pg vrg] > 0 provides a constraint for the V-representability of the g-electron reduced density matrix, or 2-matrix. Requiring that Tr[Pg Arrg] > 0 for every 2-body positive semidefinite linear operator is necessary and sufficient for the V-representability of the 2-matrix [22]. 

The constants cnn form a square (since we act on all 0 states and produce combinations of 0 states) Hermitian (since R is Hermitian) matrix in fact, cn n forms the... 

If X is the matrix formed from the eigenvectors of a matrix A, then the similarity transformation X lAX will produce a diagonal matrix whose elements are the eigenvalues of A. Furthermore, if A is Hermitian, then X will be unitary and therefore we can see that a Hermitian matrix can always be diagonalized by a unitary transformation, and a symmetric matrix by an orthogonal transformation. 

The functions (2.50) are called basis functions The matrices F, G,. .. are called matrix representatives of the operators F, G,. .. in the

specific form of the matrix representation of a set of operators depends on the basis chosen. Equation (2.53) shows that the effect of the operator G on the basis functions is determined by the matrix elements GkJ. Since an arbitrary well-behaved function can be expanded using the complete set (2.50), knowledge of the matrix G allows one to determine the effect of the operator G on an arbitrary function. Thus, knowledge of the square matrix G is fully equivalent to knowledge of the corresponding operator G. Since G is a Hermitian operator, its matrix elements satisfy Gij = (GJi). Hence the matrix G representing G is a Hermitian matrix (Section 2.1). 

The matrix obtained by taking the complex conjugate of each element of A and then forming the transpose is called the Hermitian conjugate (or conjugate transpose) of A and is symbolized by A" ... 

Matrix elements of one-body hermitian operators (such as kinetic energy, nuclear attraction, the Fock operator, etc.) have the form... 

A full proof of this can be found in Taylor [15]. The factor p is +1 if the operator is Hermitian and —1 if it is anti-Hermitian. We can see an immediate complication relative to our earlier formula Eq. 5.11 in that a full representation matrix for irrep T is required. This is considered in more detail below. Additional redundancies in the P2 list that arise from the form of particular operators axe also treated by Taylor. 

Show that a unitary matrix U can always be written in the form U = exp(T), where T is an anti-Hermitian matrix. 

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