Hermitian matrix, defined

It contains the entire description of the bath behavior in our approximation. Finally, we define a Hermitian matrix F with elements... 

The definition of irregular functions depends on boundary conditions imposed on the Green function. Any variation of the set of functions defined by adding linear combinations of the regular functions f with coefficients that constitute an Hermitian matrix simply moves the corresponding term between the two summations in Eq. (7.37), leaving the net sum invariant, and preserving the Kronecker-delta... 

From Eq. 11.5 it is apparent that ( , = p m, which defines a Hermitian matrix, so the density matrix is seen to be Hermitian. For a system of N spins /2, p is 2N X 2n in size. From the fact that the probabilities p in Eq. 11.5 must sum to unity and the basis functions used are orthonormal, it can be shown that the trace of the density matrix is 1. 

It is sometimes useful to judge the accuracy of an approximate eigenvalue or eigenvector when it is not practicable to diagonalize the matrix exactly. Consider the Hermitian matrix A and an approximate eigenvector x where x = 1. The residual vector r may be defined as... 

Since (A Af ) = (Ay, Af), the matrix x is a Hermitian matrix, so that x = X-The properties Ai,..., A define a subspace of Liouville space. This subspace is the set of all vectors that can be expressed as linear combinations of Aj,..., Am. Let us now determine the projection operator P that projects a vector onto this M-dimensional subspace. The projection operator must satisfy the conditions PA = A and P = P. Note that... 

Given a Hermitian matrix A, we can define a function of A, i.e., /(A), in much the same way we define functions f(x) of a simple variable x. For example, the square root of a matrix A, which we denote by A, is simply that matrix which when multiplied by itself gives A, i.e.,... 

We can now require that the matrix U be such that UtEU is a diagonal matrix E. (This requires that E be a hermitian matrix, which can be shown to be the case.) This requirement defines U, and we have... 

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... 

Note that on a finite-dimensional vector space V, a linear operator is Her-mitian if and only if T = T. More concretely, in C", a linear operator is Hermitian-symmetric if and only if its matrix M in the standard basis satis-lies M = M, where M denotes the conjugate transpose matrix. To check that a hnear operator is Hermitian, it suffices to check Equation 3.2 on basis vectors. Physics textbooks often contain expressions such as (+z H — z). These expressions are well defined only if H is a Hermitian operator. If H yNQK not Hermitian, the value of the expression would depend on where one applies the H. 

Exercise 3.25 Suppose M is an nyconjugate transpose of M. Suppose every eigenvalue of M is strictly positive. Define... 

Exercise 11.2 (Used in Section 11.2) Suppose M is a Hermitian-symmetric, finite-dimensional matrix (as defined in Exercise 3.25). Show that there exists a real diagonal matrix D and a unitary matrix B (see Definition 3.5) such that... 

Specifically for normal matrices, defined by the matrix equation A A = AA, this implies orthogonal diagonalizability for all normal matrices, such as symmetric (with AT = A e R" "), hermitian A = A Cn,n), orthogonal (ATA = /), unitary (A A = /), and skew-symmetric (AT = —A) matrices. 

Equations (25) and (26) show that the rows or the columns of a unitary matrix are orthonormal when the scalar product is defined to be the Hermitian scalar product. 

Consider some molecular property defined by the matrix A with the elements Ay = (( /-(l/Ur. R)l P,)) of a Hermitian operator A. One then easily derives that this matrix A obeys the following equation of motion [37],... 

For systems with high symmetry, in particular for atoms, symmetry properties can be used to reduce the matrix of the //-electron Hamiltonian to separate noninteracting blocks characterized by global symmetry quantum numbers. A particular method will be outlined here [263], to complete the discussion of basis-set expansions. A symmetry-adapted function is defined by 0 = 04>, where O is an Hermitian projection operator (O2 = O) that characterizes a particular irreducible representation of the symmetry group of the electronic Hamiltonian. Thus H commutes with O. This implies the turnover rule (0 > II 0 >) = (), which removes the projection operator from one side of the matrix element. Since the expansion of OT may run to many individual terms, this can greatly simplify formulas and computing algorithms. Matrix elements (0/x H ) simplify to (4 H v) or... 

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