Hermitian symmetric

The bracket is Hermitian symmetric. In other words, for all v,w e V we have v,w = w,vy, where the denotes complex conjugation. 

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. 

Conversely, suppose ( , > is a complex scalar product in C", Show that there is a Hermitian-symmetric matrix M such that v,w) = v Mw for any v,w e C . 

Note that every matrix AT of 5 is Hermitian symmetric, i.e., writing M to denote the conjugate transpose of M, v/e have M = M. Note also that the trace of each M e 5 is zero and... 

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... 

This is the Spectral Theorem for Hermitian-symmetric matrices. (Hint Use induction on the number of distinct eigenvalues of M.)... 

In many connections, it is convenient to introduce an abstract binary product Fl F2 associated with ordered pairs of elements of the linear space A = F, which is defined as linear in the second position, Hermitian symmetric, and positive definite ... 

By construction, all these matrices are Hermitian (symmetric in the real case) amd so, therefore, is the matrix... 

Hermicity of the Hamiltonian H implies that the coupled equations conserve probability (unitarity) even with the approximation of using a finite basis set. However, further approximations in solving the coupled equations may result in the loss of unitarity. Such loss may occur since the coupling matrix js not necessarily Hermitian symmetric. Note that H is not equal to HL, in general. A well-known method to ensure unitarity is to force the coupling matrix to be Hermitian symmetric by... 

Although the matrix (H ) is not Hermitian symmetric, unitarity is retained when the coupled equations (20) are solved exactly. This has been done numerically by Lin and collaborators using wave functions with translational factors and realistic interaction potentials. Similarly, Pfeiffer and Garcia applied nonHermitian symmetric matrix elements under conditions preserving unitarity. They have been able to solve the coupled equations analytically on the basis of hydrogenic orbitals. However, in the calculation the overlap of the wave functions and the distortion in the diabatic potential have been neglected. 

Orthogonalized basis states have been used in the calculations by Fritsch et and Stolterfoht. Moreover, Hermitian symmetric matrix elements have been achieved by Pfeiffer and Garcia using the arithmetic averaging procedure. Moreover, Hermitian symmetry is assumed for the analytic model matrix element. ... 

Let [A] be an n X n matrix (and Hermitian symmetric for quantum mechanics) with eigenvectors... 

Such a matrix is said to be Hermitian-symmetric. The concise expression on the right-hand side of (2.2.3) is evidently a row times a square matrix times a column (all conformable), yielding a single number (a 1 x 1 matrix). The space defined by an infinite set of functions i with a metric defined by (2.2.4), and with further properties to be described, is called a Hilbert space. 

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