Hermitian complex matrix

The simplest way to diagonalize the Hermitian complex matrix S(k) defined by equation (1.19) is to rewrite its eigenvalue equation (1.25) in the form... 

The operation of complex conjugation will be denoted by an overscore a denotes the complex conjugate of a. For a matrix A, with matrix elements al the hermitian conjugate matrix with elements afl will be denoted by A ... 

This number is the answer to the question originally posed. This is the number of real conditions required to fix experimentally a complex, normalized, hermitian, projection matrix. For example, this number of experimental structure factors, Equation (1), would suffice to fix P Equation (6). 

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 . 

In addition to the Hermitian polarization matrix with complex elements, the spatial anisotrophy of the electromagnetic field can be described by an equivalent set of real Stokes parameters [14,57]. 

It is convenient, for simple systems, to have explicit expressions for equation (B2.4.17). Since the original matrix is non-Hermitian, the matrix formed by the eigenvectors will not be unitary, and will have four independent complex elements. Let them be a, b, c and d, so that U is given by equation (B2.4.20). 

Specifically, by diagonalizing an appropriately constructed non-Hermitian Hamiltonian matrix and by producing the final solution in the form of an expansion over configurations with complex coefficients,... 

As regards the theoretical framework, the form of the resonance eigenfunction according to Eq. (1) constitutes a reference point. Eq. (1) is related to time-dependent analysis (Eq. (6) and subsequent discussion) or to Hermitian K-matrix computations that depend on real values of the energy (Sections 3.1.1 and 3.1.2 and Chapter 6) or to computations that depend on complex energies and non-Hermitian constructions (Sections 3-7, 11). As regards the computational framework, Eq. (1), is implemented in terms of wavefunctions of the form of Eqs. (32,35,37, and 48). 

For each TCNQ and TTF chain the eigenvalue problem of a complex Hermitian matrix of order 68 and 64, respectively, had to be solved 80-150 times. The calculations were accelerated by diagonalizing the Hermitian complex matrices with the aid of a complex matrix eigenvalue program based on the QR algorithm.< > The resulting computing time was only 6-9 min for one chain on the IBM 360/91 computer. 

It seems the key step in this derivation, which differs from the analysis of CGM, is the following. In the system of equations resulting from the constraint C C+ = Ijv, Pecora considers that N(N - 1) of [them] are simply complex conjugates of each other , yielding a total number of complex conditions equal to N(N + l)/2. This is, in fact, equivalent to considering theCC1 matrix as hermitian, i.e.,... 

However, if one were to exactly follow what seem to be Pecora s assumptions about the scalar product being hermitian, one would get a different result from Pecora when counting the number of real conditions on the complex P matrix, arising from the constraint + = In In fact, when the + matrix is considered to be hermitian, the normalization condition on the N complex diagonal elements of QQ+ yields N real conditions and not 2N as Pecora seemed to tacitly suppose. This is due to the fact that the diagonal elements are already known to be real since + is hermitian, and hence, Im = 0 is not a separate constraint. 

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. 

These coupling matrix elements are scalars due to the presence of the scalar Laplacian in Eq. (25). These elements are, in general, complex but if we require the /)L id to be real they become real. The matrix Wl-2 ad(R-/.), unlike its first-derivative counterpart, is neither skew-Hermitian nor skew-symmetric. 

This matrix is the appropriate representation of an observable such as X. A Hermitian matrix is its own hermitian conjugate. The diagonal elements of a Hermitian matrix are real and each element is symmetry related to its complex conjugate across the main diagonal. 

Because all quantum-mechanical operators are Hermitian, the corresponding matrices are also Hermitian. In other words, the complex conjugate of the transpose of such a matrix (denoted as is equal to itself ... 

HOC1,309,310 HArF,311 and C1HC1.71 Most of these calculations were carried out using either the complex-symmetric Lanczos algorithm or filter-diagonali-zation based on the damped Chebyshev recursion. The convergence behavior of these two algorithms is typically much less favorable than in Hermitian cases because the matrix is complex symmetric. 

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... 

Note that the Liouville matrix, iL + R + K may not be Hermitian, but it can still be diagonalized. Its eigenvalues and eigenvectors are not necessarily real, however, and the inverse of U may not be its complex-conjugate transpose. If we allow complex numbers in the equation, (12) is a general result. Since A is a diagonal matrix we can expand in terms of the individual eigenvalues, Xj. We can also apply U ( backwards )... 

The quantity f g is called the inner product (or scalar product) of the column vectors f and g. The inner product is a generalization of the dot product (1.55) to vectors with an arbitrary number of complex components. Since a Hermitian operator satisfies (1.13), then (2.63) shows that for a Hermitian matrix A... 

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" ... 

Prove that ftAtg = gA f > where f and g are n by 1 column vectors and A is an n by n square matrix. [When A is Hermitian, this identity reduces to the equation following (2.63).] Hint take the Hermitian conjugate of gtAf, noting that this product is a complex scalar. 

Let us recall that the transformation matrix D1 is the Hermitian conjugate to matrix D and obtained from D by its transposition (D) and complex conjugation ( > ). The unitarity of the matrix implies that... 

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