Hermitian-symmetric matrix

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 . 

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. ... 

It is well established that the eigenvalues of an Hermitian matrix are all real, and their corresponding eigenvectors can be made orthonormal. A special case arises when the elements of the Hermitian matrix A are real, which can be achieved by using real basis functions. Under such circumstances, the Hermitian matrix is reduced to a real-symmetric matrix ... 

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. 

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... 

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. 

State whether each of the following concepts is applicable to all matrices or to only square matrices (a) real matrix (b) symmetric matrix (c) diagonal matrix (d) null matrix (e) unit matrix (f) Hermitian matrix (g) orthogonal matrix (h) transpose (i) inverse (j) Hermitian conjugate (k) eigenvalues. 

The diagonal elements of a Hermitian matrix must be real. A real symmetric matrix is a special case of a Hermitian matrix. (The relation between Hermitian matrices and Hermitian operators will be shown in Section 2.3.)... 

Note that a symmetric matrix is unchanged by rotation about its principal diagonal. The complex-number analogue of a symmetric matrix is a Hermitian matrix (after the mathematician Charles Hermite) this has atJ = a, e.g. if element (2,3) = a + bi, then element (3,2) = a — bi, the complex conjugate of element (2,3) i = f 1. Since all the matrices we will use are real rather than complex, attention has been focussed on real matrices here. 

Now, it can be proved that if and only if A is a symmetric matrix (or more generally, if we are using complex numbers, a Hermitian matrix - see symmetric matrices, above), then P is orthogonal (or more generally, unitary - see orthogonal matrices, above) and so the inverse P 1 of the premultiplying matrix P is simply the... 

Since He is Hermitian the matrix is a real symmetric matrix. This N x N determinant gives an ATh-order polynomial in E. The lowest root is the best approximation to the ground-state energy within the framework of the trial function in Eq. (3.7). 

By the Hellmann-Feynman theorem [46], the derivative of an eigenvalue Ilgu (s) of a hermitian (here, real symmetric) matrix II11 (.y) with respect to a parameter h[ is given by the diagonal element of the matrix product,... 

Bap being a hermitian matrix, and Aap a symmetric matrix. According to the Tyablikov-Bogoljubov method66 the diagonalization of the quadratic form (A.l) can be performed by introducing new operators... 

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

There is a separate solution corresponding to each possible incoming channel, and the solution is characterized at long range by the S-matrix with elements Sji. The S-matrix is an A open x A open complex symmetric matrix, where A open is the number of open channels. It is unitary, that is, SS = I, where indicates the Hermitian conjugate and / is a unit matrix. If the physical problem is factorized into separate sets of coupled equations for different symmetries (such as total angular momentum or parity), there is a separate S-matrix for each symmetry. All properties that correspond to completed collisions, such as elastic and inelastic integral and differential cross-sections, can be written in terms of S-matrices. 

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. 

A square matrix B is a symntetric matrix if aU its elements satisfy b = b m- The elements of a symmetric matrix are symmetric about the principal diagonal for example, b 2 = 21- A square matrix D is a Hermitian matrix if all its elements satisfy 4 = d -For example, if... 

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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