Diagonalization Problem Research Articles

Algebraic techniques for diagonalization of a split quaternion matrix in split quaternionic mechanics

In the study of the relation between complexified classical and non-Hermitian quantum mechanics, physicists found that there are links to quaternionic and split quaternionic mechanics, and this leads to the possibility of employing algebraic techniques of split quaternions to tackle some problems in complexified classical and quantum mechanics. This paper, by means of real representation of a split quaternion matrix, studies the problem of diagonalization of a split quaternion matrix and gives algebraic techniques for diagonalization of split quaternion matrices in split quaternionic mechanics.

Some perturbation results for a normalized Non-Orthogonal Joint Diagonalization problem

Non-Orthogonal Joint Diagonalization (NOJD) of a given real symmetric matrix set A={Aj}j=0p is to find a nonsingular matrix W such that W⊤AjW for j=0,1,…,p are all as diagonal as possible. If the columns of the solution W are all required to be unit length, we call such NOJD problem as the Normalized NOJD (NNOJD) problem. In this paper, we discuss the perturbation theory for NNOJD as an optimization problem. Based on the perturbation analysis of general constrained optimization problem given in [16], we obtain an upper bound for the distance between an approximated solution of the perturbed optimal problem and the set of exact joint diagonalizers. As corollaries, a perturbation bound and an error bound are also given. Numerical examples validate the bounds.

Relative gradient based algorithms for general joint diagonalization of complex matrices

This article deals with the problem of joint diagonalization of hermitian and/or complex symmetric matrices. Within the framework of gradient algorithms, we develop various algorithms which are based on different levels of approximation of the classical diagonalization criterion. The algorithms are based on a multiplicative update and on the derivation of an optimal step-size. One of the algorithms is a generalization of DOMUNG to the complex case. Finally, in the blind source separation context, computer simulations illustrate the relative performances of some proposed algorithms in comparison to the true gradient one.

A Decoupled Jacobi-Like Algorithm for Non-Unitary Joint Diagonalization of Complex-Valued Matrices

We consider the problem of non-orthogonal joint diagonalization of a set of complex matrices. This appears in many signal processing problems and is instrumental in source separation. We propose a new Jacobi-like algorithm based both on a special parameterization of the diagonalizing matrix and on an adapted local criterion. The optimization scheme is based on an alternate estimation of the useful parameters. Numerical simulations illustrate the overall very good performances of the proposed algorithm in comparison to two other Jacobi-like algorithms and to a global algorithm existing in the literature.

A Coordinate Descent Algorithm for Complex Joint Diagonalization Under Hermitian and Transpose Congruences

This paper deals with the problem of joint complex matrix diagonalization by considering both the Hermitian and transpose congruences. We address the general case where the searched diagonalizing matrix is a priori nonunitary. Based on the minimization of a quadratic criterion, we propose a flexible algorithm in the sense that it allows to directly consider a rectangular diagonalizing matrix and to take into consideration both the Hermitian and transpose congruences within the same framework. The proposed algorithm is a coordinate descent algorithm that is based on an approximate criterion leading to the analytical derivation of the minima arguments. Computer simulations are drawn to illustrate the usefulness and performances of the algorithm and a comparison to state-of-the-art algorithms is presented. Finally, an application to independent component analysis based on fourth-order statistics is also presented.

Constrained optimization of the overdetermined Zimmermann compressor for nonorthogonal joint matrix diagonalization

The classical $$2 \times 2$$ 2 × 2 Zimmermann matrix known from nonorthogonal Jacobi algorithms for symmetric definite matrix pairs can be used as an elementary compressor in the overdetermined problem of joint matrix diagonalization of more than two matrices. For this purpose, a constrained optimization procedure for adapting the overdetermined Zimmermann compressor to offdiagonal error minimization is established. It turns out that in the case of the Zimmermann matrix, the constrained optimization approach exhibits a key feature in terms of a $$3 \times 3$$ 3 × 3 rank-deficient constraint matrix. This rank drop simplifies the complexity of Zimmermann matrix design relative to other possible types of compressor matrices in this application. A minimum complexity nonorthogonal Jacobi algorithm for joint matrix diagonalization is hence obtained on the basis of the overdetermined Zimmermann compressor. We show diagonalization results obtained with this algorithm for extreme cases like the joint diagonalization of only $$3$$ 3 highly-correlated matrices of dimension $$500 \times 500$$ 500 × 500 . A "null space" variant of the classical Zimmermann algorithm for matrix pairs is obtained as a by-product from the overdetermined joint matrix diagonalizer.

A Riemannian symmetric rank-one trust-region method

The well-known symmetric rank-one trust-region method--where the Hessian approximation is generated by the symmetric rank-one update--is generalized to the problem of minimizing a real-valued function over a $$d$$ d -dimensional Riemannian manifold. The generalization relies on basic differential-geometric concepts, such as tangent spaces, Riemannian metrics, and the Riemannian gradient, as well as on the more recent notions of (first-order) retraction and vector transport. The new method, called RTR-SR1, is shown to converge globally and $$d+1$$ d + 1 -step q-superlinearly to stationary points of the objective function. A limited-memory version, referred to as LRTR-SR1, is also introduced. In this context, novel efficient strategies are presented to construct a vector transport on a submanifold of a Euclidean space. Numerical experiments--Rayleigh quotient minimization on the sphere and a joint diagonalization problem on the Stiefel manifold--illustrate the value of the new methods.

A New Algorithm for Complex Non-Orthogonal Joint Diagonalization Based on Shear and Givens Rotations

This paper introduces a new algorithm to approximate non-orthogonal joint diagonalization (NOJD) of a set of complex matrices. This algorithm is based on the Frobenius norm formulation of the joint diagonalization (JD) problem and takes advantage from combining Givens and Shear rotations to attempt the approximate JD. It represents a non-trivial generalization of the JDi (Joint Diagonalization) algorithm (Souloumiac 2009) to the complex case. The JDi is first slightly modified then generalized to the CJDi (i.e., Complex JDi) using complex to real matrix transformation. Several methods already exist in the literature. Consequently, we provide herein a brief overview of existing NOJD algorithms. Then we provide an extensive comparative study to illustrate the effectiveness and stability of the CJDi with respect to various system parameters and application contexts.

Structural characterization and condition for measurement statistics preservation of a unital quantum operation

We investigate the necessary and sufficient condition for a convex cone of positive semidefinite operators to be fixed by a unital quantum operation ϕ acting on finite-dimensional quantum states. By reducing this problem to the problem of simultaneous diagonalization of the Kraus operators associated with ϕ, we can completely characterize the kinds of quantum states that are fixed by ϕ. Our work has several applications. It gives a simple proof of the structural characterization of a unital quantum operation that acts on finite-dimensional quantum states—a result not explicitly mentioned in earlier studies. It also provides a necessary and sufficient condition for determining what kind of measurement statistics is preserved by a unital quantum operation. Finally, our result clarifies and extends the work of Størmer by giving a proof of a reduction theorem on the unassisted and entanglement-assisted classical capacities, coherent information, and minimal output Renyi entropy of a unital channel acting on a finite-dimensional quantum state.

Algebraic Methods for Condiagonalization Under Consimilarity of Quaternion Matrices in Quaternionic Quantum Mechanics

By means of complex representation and real representation of quaternion matrices, paper [7] studied the problem of diagonalization of quaternion matrices. This paper introduces two new complex representation and real representation of quaternion matrices, studies the problem of condiagonalization under consimilarity of quaternion matrices, and gives two algebraic methods for the condiagonalization under consimilarity of quaternion matrices.

Dynamics of semiflexible regular hyperbranched polymers

We study the dynamics of semiflexible Vicsek fractals (SVF) following the framework established by Dolgushev and Blumen [J. Chem. Phys. 131, 044905 (2009)], a scheme which allows to model semiflexible treelike polymers of arbitrary architecture. We show, extending the methods used in the treatment of semiflexible dendrimers by Fürstenberg et al. [J. Chem. Phys. 136, 154904 (2012)], that in this way the Langevin-dynamics of SVF can be treated to a large part analytically. For this we show for arbitrary Vicsek fractals (VF) how to construct complete sets of eigenvectors; these reduce considerably the diagonalization problem of the corresponding equations of motion. In fact, such eigenvector sets arise naturally from a hierarchical procedure which follows the iterative construction of the VF. We use the obtained eigenvalues to calculate the loss moduli G(")(ω) of SVF for different degrees of stiffness of the junctions. Finally, we compare the results for SVF to those found for semiflexible dendrimers.

Progressive Linear Precoder Optimization for ARQ Packet Retransmissions in Nonregenerative MIMO Relay Systems

This paper investigates the linear precoder optimization for packet retransmissions in nonregenerative multi-input multi-output (MIMO) relay systems. To fully utilize the time diversity provided by automatic repeat request (ARQ), progressive linear precoders are designed with the objective of maximizing the mutual information delivered by multiple transmissions of the same packet. Firstly, the relay-only precoder is considered when the channel state information (CSI) is only available at the relay and the destination. We derive the form of the optimal ARQ precoder by restating the optimization problem as a matrix diagonalization problem, then the power allocation is formulated as a convex optimization problem where the Karush-Kuhn-Tucker conditions are used to obtain the optimal solution. Furthermore, we analyze the asymptotic performance when the two hops' signal-to-noise ratio (SNR) goes to infinity or zero respectively. Secondly, the joint source and relay precoder is studied assuming the CSI known at all nodes. The ARQ precoder is derived by matrix diagonalization and channel pairing between previous subchannels and current subchannels. Then the power allocation at the source and relay is formulated as a nonconvex multi-variable optimization problem where we employ an iterative algorithm to obtain the suboptimal solution. Moreover, the asymptotic performance is also analyzed with the two hops' SNR being extremely large or small, based on which the simplified method of searching channel pairing matrix is proposed. Numerical results show that the proposed precoders can improve the system performance significantly, especially in the joint source and relay precoder scenario.

On the rhotrix eigenvalues and eigenvectors

The concept of rhotrix eigenvector eigenvalue problem (REP) was introduced by Aminu (Int. J. Math. Educ. Sci. Technol. 41:98–105, 2010). As an extension to this, we have presented in this article some properties of rhotrix eigenvalues and eigenvectors considering the numerous applications of matrix eigenvector eigenvalue problem in areas of Applied Mathematics. A classic problem in linear algebra, the diagonalization problem in terms of rhotrices (RDP) is also introduced.

A “Sequentially Drilled” Joint Congruence (SeDJoCo) Transformation With Applications in Blind Source Separation and Multiuser MIMO Systems

We consider a particular form of the classical approximate joint diagonalization (AJD) problem, which we call a “sequentially drilled” joint congruence (SeDJoCo) transformation. The problem consists of a set of symmetric real-valued (or Hermitian-symmetric complex-valued) target-matrices. The number of matrices in the set equals their dimension, and the joint diagonality criterion requires that in each transformed (“diagonalized”) target-matrix, all off-diagonal elements on one specific row and column (corresponding to the matrix-index in the set) be exactly zeros, yet does not care about the other (diagonal or off-diagonal) elements. The motivation for this form arises in (at least) two different contexts: maximum likelihood blind (or semiblind) source separation and coordinated beamforming for multiple-input multiple-output (MIMO) broadcast channels. We prove that SeDJoCo always has a solution when the target-matrices are positive-definite . We also propose two possible iterative solution algorithms, based on defining and optimizing two different criteria functions, using Newton's method for the first function and successive Jacobi-like transformations for the second. The algorithms' convergence behavior and the attainable performance in the two contexts above are demonstrated in simulation experiments.

Fast nonunitary joint block diagonalization with degenerate solution elimination for convolutive blind source separation

This paper addresses the problem of joint block diagonalization (JBD) of a set of given matrices. As is known that the nonunitary JBD algorithm has some advantages over the existing orthogonal one for convolutive blind source separation (CBSS). However, the nonunitary JBD algorithm is prone to converge to some unexpected degenerate solutions (singular or ill-conditioned solutions). Especially for the matrices of large dimension or the case that the number of the diagonal blocks is relatively large, the performances of the nonunitary JBD algorithm degrade more severely. To eliminate the degenerate solutions, we optimize a penalty term based weighted least-squares criterion and thus develop a fast efficient algorithm. The performance of the proposed algorithm is evaluated by computer simulations and compared with the existing state-of-the-art nonunitary JBD algorithm. The simulation results demonstrate the robustness and performance improvement of the proposed algorithm.

Response calculations based on an independent particle system with the exact one-particle density matrix: Excitation energies

Adiabatic response time-dependent density functional theory (TDDFT) suffers from the restriction to basically an occupied → virtual single excitation formulation. Adiabatic time-dependent density matrix functional theory allows to break away from this restriction. Problematic excitations for TDDFT, viz. bonding-antibonding, double, charge transfer, and higher excitations, are calculated along the bond-dissociation coordinate of the prototype molecules H(2) and HeH(+) using the recently developed adiabatic linear response phase-including (PI) natural orbital theory (PINO). The possibility to systematically increase the scope of the calculation from excitations out of (strongly) occupied into weakly occupied ("virtual") natural orbitals to larger ranges of excitations is explored. The quality of the PINO response calculations is already much improved over TDDFT even when the severest restriction is made, to virtually the size of the TDDFT diagonalization problem (only single excitation out of occupied orbitals plus all diagonal doubles). Further marked improvement is obtained with moderate extension to allow for excitation out of the lumo and lumo+1, which become fractionally occupied in particular at longer distances due to left-right correlation effects. In the second place the interpretation of density matrix response calculations is elucidated. The one-particle reduced density matrix response for an excitation is related to the transition density matrix to the corresponding excited state. The interpretation of the transition density matrix in terms of the familiar excitation character (single excitations, double excitations of various types, etc.) is detailed. The adiabatic PINO theory is shown to successfully resolve the problematic cases of adiabatic TDDFT when it uses a proper PI orbital functional such as the PILS functional.

A novel joint diagonalization approach for linear stochastic systems and reliability analysis

PurposeThe purpose of this paper is to present a novel gradient‐based iterative algorithm for the joint diagonalization of a set of real symmetric matrices. The approximate joint diagonalization of a set of matrices is an important tool for solving stochastic linear equations. As an application, reliability analysis of structures by using the stochastic finite element analysis based on the joint diagonalization approach is also introduced in this paper, and it provides useful references to practical engineers.Design/methodology/approachBy starting with a least squares (LS) criterion, the authors obtain a classical nonlinear cost‐function and transfer the joint diagonalization problem into a least squares like minimization problem. A gradient method for minimizing such a cost function is derived and tested against other techniques in engineering applications.FindingsA novel approach is presented for joint diagonalization for a set of real symmetric matrices. The new algorithm works on the numerical gradient base, and solves the problem with iterations. Demonstrated by examples, the new algorithm shows the merits of simplicity, effectiveness, and computational efficiency.Originality/valueA novel algorithm for joint diagonalization of real symmetric matrices is presented in this paper. The new algorithm is based on the least squares criterion, and it iteratively searches for the optimal transformation matrix based on the gradient of the cost function, which can be computed in a closed form. Numerical examples show that the new algorithm is efficient and robust. The new algorithm is applied in conjunction with stochastic finite element methods, and very promising results are observed which match very well with the Monte Carlo method, but with higher computational efficiency. The new method is also tested in the context of structural reliability analysis. The reliability index obtained with the joint diagonalization approach is compared with the conventional Hasofer Lind algorithm, and again good agreement is achieved.

A Hybrid Approach to the Non-orthogonal Joint Diagonalization Problem(Application)

A Hybrid Approach to the Non-orthogonal Joint Diagonalization Problem(Application)

Use of Gauss-Hermite quadrature to approximate the asymptotic behavior of vibronic resonance wave functions

Wave functions obtained employing a standard complex Hamiltonian matrix diagonalization procedure are square-integrable and therefore cannot accurately describe the asymptotic character of resonance solutions of the Schrodinger equation. The nature of this approximation is investigated by means of explicit calculations which are based on diabatic RKR potentials for the B 1Σ+ — D′ 1Σ+ vibronic resonance states of the CO molecule. It is shown that expanding the basis of complex harmonic oscillator fuctions gradually improves the description of the exact resonance wave functions out to ever larger internuclear distances on the real axis before they take on artificial bound-state characteristics due to the square-integrable character of the basis functions. In order to solve the diagonalization problems for as many as 500 such basis functions, it proves necessary to employ specialized numerical techniques such as Gauss-Hermite quadrature to evaluate the required Hamiltonian matrix elements.

A new discriminant subspace analysis approach for multi-class problems

Fukunaga-Koontz Transform (FKT) is a famous feature extraction method in statistical pattern recognition, which aims to find a set of vectors that have the best representative power for one class while the poorest representative power for the other class. Li and Savvides [1] propose a one-against-all strategy to deal with multi-class problems, in which the two-class FKT method can be directly applied to find the presentative vectors of each class. Motivated by the FKT method, in this paper we propose a new discriminant subspace analysis (DSA) method for the multi-class feature extraction problems. To solve DSA, we propose an iterative algorithm for the joint diagonalization (JD) problem. Finally, we generalize the linear DSA method to handle nonlinear feature extraction problems via the kernel trick. To demonstrate the effectiveness of the proposed method for pattern recognition problems, we conduct extensive experiments on real data sets and show that the proposed method outperforms most commonly used feature extraction methods.