Complex-symmetric matrices

Without losing generality, this review will concentrate on real-symmetric matrices, whereas their Hermitian counterparts can be handled in a similar way. In some special cases, solutions of complex-symmetric matrices are required. This situation will be discussed separately. 

Complex-symmetric matrices might arise in some problems in chemical physics. Examples include the electron paramagnetic resonance (EPR) and nuclear magnetic resonance (NMR) line shape problems.207 Another... 

C.E. Reid, E. Brandas, On a Theorem for Complex Symmetric Matrices and Its Relevance in the Study of Decay Phenomena, Lect. Notes. Phys. 325 (1989) 476. 

E. Brandas, A Theorem for Complex Symmetric Matrices Revisited, Int. J. Quant. Chem. 109 (2009) 2860. 

The use of complex coordinates either in the Hamiltonian or in the trial functions leads to complex symmetric matrices, which are often large due to the necessity of employing large basis. Some effective algorithms cind computer codes have been recently developed (130-132) for diagonalizing such matrices. 

The application of the Chebyshev recursion to complex-symmetric problems is more restricted because Chebyshev polynomials may diverge outside the real axis. Nevertheless, eigenvalues of a complex-symmetric matrix that are close to the real energy axis can be obtained using the FD method based on the damped Chebyshev recursion.155,215 For broad and even overlapping resonances, it has been shown that the use of multiple cross-correlation functions may be beneficial.216... 

First we observe that any matrix is similar to a block diagonal matrix, where the sub-matrices along the main diagonal are Jordan blocks. It is thus sufficient to prove that any Jordan block can be transformed to a complex symmetric matrix. In passing we note that any matrix with distinct eigenvalues can be brought to diagonal form by a similarity transformation. The key study therefore relates to XI + J (0), where 1 is the n-dimensional unit matrix and... 

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. 

Coefficients and actually form the first and second diagonal of the tridiagonal (complex) symmetric matrix representation of the symmetrized Liouvillian, and the spectrum can be written in the form of a continued fraction [84]... 

We note that the BIP is the inner product for which the eigenvectors of a complex symmetric matrix are orthogonal, which is the mathematical grounding for its use. With this modification, the Lanczos recursion formula becomes... 

From the secular equation, based on the complex symmetric matrix 3 [, we obtain the eigenvalues X = mo via the Klein-Gordon-like equation. 

The symmetry argument actually goes beyond the above deterniination of the symmetries of Jahn-Teller active modes, the coefficients of the matrix element expansions in different coordinates are also symmetry determined. Consider, for simplicity, an electronic state of symmetiy in an even-electron molecule with a single threefold axis of symmetry, and choose a representation in which two complex electronic components, e ) = 1/v ( ca) i cb)), and two degenerate complex nuclear coordinate combinations Q = re " each have character T under the C3 operation, where x — The bras e have character x. Since the Hamiltonian operator is totally symmetric, the diagonal matrix elements e H e ) are totally symmetric, while the characters of the off-diagonal elements ezf H e ) are x. Since x = 1, it follows that an expansion of the complex Hamiltonian matrix to quadratic terms in Q. takes the form... 

The Hermetian conjugate plays the same role for complex matrices that the symmetric matrix plays for real matrices. 

The matrices [G] and [F] are column matrices with row numbers n and k, respectively. The matrix solution is simplified by special properties of the symmetric matrix and because the resulting values of G occur in complex conjugate pairs. In general, we may write... 

If a matrix is equal to its transpose, it is said to be a symmetric matrix. If the elements of A are complex numbers, the complex conjugate of A is defined as... 

A.7 Show that for a 2 X 2 symmetrical matrix, the eigenvalues must be real (do not contain imaginary components). Develop a 2 X 2 nonsymmetrical matrix which has complex eigenvalues. 

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. 

Method for Calculating Transition Amplitudes. III. S-Matrix Elements with a Complex-Symmetric Hamiltonian. 

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 . 

The determinant of A is nonzero if A is nonsingular, so the solutions to the two detenninantal equations must be the same. B A is the inverse of A 1B, so its characteristic roots must be the reciprocals of those of A" B. There might seem to be a problem here since these two matrices need not be symmetric, so the roots could be complex. But, for the application noted, both A and B are symmetric and positive definite. As such, it can be shown (see Section 16.5.2d) that the solution is the same as that of a third determinantal equation involving a symmetric matrix. 

For convenience we will make a simple demonstration of how to transform a 2x2 matrix problem to complex symmetric form. In so doing we will also recognise the appearence of a Jordan block off the real axis as an immediate consequence of the generalisation. The example referred to is treated in some detail in Ref. [15], where in addition to the presence of complex eigenvalues one also demonstrates the crossing relations on and off the real axis. The Hamiltonian... 

Equation (97) serves as a boundary condition for the operator matrix model, which in the complex symmetric representation, making the replacement v/c = k(t), reads (note that x(r) < b see below)... 

For our example we assume, for more relevant details see further below, that the operator W(r) and the functions y(r), e/>(r) are aptly defined for the scaling to be meaningful. Such a family of potentials will be denoted by jH0 [53]. If the family has an extension, see more details below, to arg r] < 0 analytic in the interior and up to the boundary, the class is denoted by jy0. At this point, it is trivial to deduce from Eq. (D.l) that the matrix element is analytic in the parameter r] provided that the complex conjugate of i] is inserted on the "bra" side of Eq. (D.2). Hence, many complex scaling treatments in quantum chemistry are operationally derived from complex symmetric representations. 

Theorem Any matrix can be brought to complex symmetric form by a similarity transformation. 

As already stated, the transformation B is not unique. Nonetheless it is interesting, with the aforementioned intermezzo as background, to note that the result (E.13), to be demonstrated below, i.e., em(k+l 2)/n(Ski — 1) carries a crucial semblance with r 2) in Eq. (E.5). In fact T(2), see more below, relates the complex symmetric representation of the Jordan block through thermaliza-tion, and furthermore the matrix B can also be used to diagonalize the latter. Employing the transformation H)B = g) = g, g2, g ) Eq. (E.5) writes... 

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... 

The derivation given above of the stationary Kohn functional [ K] depends on logic that is not changed if the functions Fo and l< of Eq. (8.5) are replaced in each channel by any functions for which the Wronskian condition mm — m 0 = l is satisfied [245, 191]. The complex Kohn method [244, 237, 440] exploits this fact by defining continuum basis functions consistent with the canonical form cv() = I.a = T, where T is the complex-symmetric multichannel transition matrix. These continuum basis functions have the asymptotic forms... 

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