Hermitian tridiagonal solution with the least norm to quaternionic least squares problem☆
Introduction
Quaternions and quaternion matrices play essential roles in quaternionic quantum mechanics and field theory [1], [2], [3], [4], [5], [6], [7], [8]. In the study of theory and numerical computations of quaternionic quantum theory, in order to well understand the perturbation theory [1], experimental proposals [3], [4], [5] and theoretical discussions [6], [7], [8] underlying the quaternionic formulations of energy eigenvalue equation and so on, one often meets problems that finding approximate solutions of quaternion linear systems, such as the approximate solution of quaternion linear matrix equation that is approximate when error occurs in the observation matrix E, i.e. quaternionic least squares (QLS) problem, which has received lots of attention in mathematics and physics recently [9], [10], [11], [12], [13], [14].
As is well known, the main difficulty dealing with the QLS problem lies in the non-commutation of the multiplicity of quaternions, thus the general methods for least squares (LS) problem in real number field or complex number field can't be directly used. By means of complex representation, T. Jiang studied algebraic algorithms for QLS problem [10], QLS eigenproblem [11] and QLS problem with constraints [12], [13] in quaternionic quantum theory, and obtained some theoretical results. In [14], the QLS problem in quaternionic quantum theory was reconsidered by real representation, taking advantage of structures of the real representation matrices, the authors derived an operable iterative method called LSQR-Q algorithm for finding the minimum-norm solution of the QLS problem, which is more appropriate to large scale system. To our knowledge, Hermitian tridiagonal solution with the least norm to the QLS problem has not been investigated so far in the literature.
In real number filed or complex number field, many problems in physical applications, such as heat exchange, diffusing process and structure analyses, etc., give rise to a linear system whose coefficient matrix is symmetric tridiagonal or Hermitian tridiagonal. As fundamental tools, symmetric (Hermitian) tridiagonal matrices were widely studied and applied in distinct points of view [15], [16], [17], [18], [19]. In [15], analytical formula for the inversion of symmetrical tridiagonal matrices was presented which could be used to derive an exact analytical solution for the one-dimensional discrete Poisson equation with Dirichlet boundary conditions, paper [17] obtained exact energy level correlators for the Gaussian ensemble of finite tridiagonal symmetric matrices. By relaxing the diagonal constraint on the matrix representation of eigenvalue wave equation to tridiagonal form, a closed form solution of the three-dimensional time-independent Schrödinger equation for a charged particle in the field of a point electric dipole that could carry a nonzero net charge was obtained [19].
Motivated by the work of [14], and keeping the applications and interests of tridiagonal matrix in physical applications mentioned above, in this paper we consider the Hermitian tridiagonal solution with the least norm to the QLS problem in quaternionic quantum theory.
Let R be the real number field, C the complex number field, the quaternion field, where , . denotes the set of matrices over field F, denotes the set of tridiagonal matrices of order n. For any quaternion where , , the conjugate of quaternion x is and . For any matrix , , , denote the transpose, conjugate transpose and Moore–Penrose inverse of A, respectively. represents the submatrix of A containing the intersection of rows i to j and columns k to l. As a special case, is the ith to jth columns of A and the ith to jth rows of A. denotes the Kronecker product of two matrices A and B, the ith column of identity matrix with appropriate size.
Given a tridiagonal matrix with the form if and , then A is a symmetric tridiagonal matrix, if and , then A is a skew-symmetric tridiagonal matrix, if and , then A is a Hermitian tridiagonal matrix, we denote the set of symmetric tridiagonal matrix, skew-symmetric tridiagonal matrix and Hermitian tridiagonal quaternion matrix by , , , respectively. It is not difficult to verify that if , then .
For any quaternion matrix , the Frobenius norm of quaternion matrix is defined in [14] to be where is the real representation matrix of A with the form the set of all matrices shaped like (1.4) is denoted by and is the Frobenius norm.
We consider the following quaternionic least squares (QLS) problem. For given data matrices , , an observation matrix , write
we are asked to find a matrix such that
Section snippets
Basis matrices and subvectors
In view of the special structure of Hermitian tridiagonal quaternion matrices, we construct basis matrices and subvectors, study the relationship between Hermitian tridiagonal matrices and basis matrices and subvectors in this section. For more discusses we refer to Appendix A at the end of the paper.
For any matrix , define and are
Algorithm for the QLS problem (1.5)–(1.6)
In this section, we study the QLS problem mentioned in the first section by the LSQR-Q algorithm. In view of the special structure of Hermitian tridiagonal quaternion matrix, our main task lies in constructing constrained matrices in the algorithm.
Let
Lemma 3.1 (See [14].) For any , is a real representation matrix of some quaternion matrix.
For any matrix ,
Preconditioning
Wiegmann [21] gave the following singular value decomposition (SVD) of a quaternion matrix.
Theorem 4.1 (See [21].) Let with . Then there exist unitary quaternion matrices and such that where , , and are the nonzero singular values of quaternion matrix A.
Taking the real representation operation in both sides of (4.1), we get a SVD of real representation matrix It follows that the singular values of
Numerical experiments
In this section, we first verify the effectiveness of our Algorithm 3.1, then we illustrate that our preconditioning strategy is efficient by two examples. The experiments are performed by MATLAB 7.3.0 on PC machine (Intel Celeron 2.66 GHz, Memory 512 MB), all functions are defined by MATLAB 7.3.0.
Example 5.1 Given , , , . Take , , , , , . Let where , M,
Acknowledgements
The authors are indebted to the anonymous referee and the editor for their helpful comments and suggestions, which improved the presentation of this paper.
References (22)
Schrödinger inviolate: Neutron optical searches for violations of quantum mechanics
Physica B
(1988)- et al.
Algebraic algorithms for least squares problem in quaternionic quantum theory
Comput. Phys. Commun.
(2007) - et al.
An algebraic method for Schrödinger equations in quaternionic quantum mechanics
Comput. Phys. Commun.
(2008) - et al.
A new technique of quaternion equality constrained least squares problem
J. Comput. Appl. Math.
(2008) - et al.
Equality constrained least squares least problem over quaternion filed
Appl. Math. Lett.
(2003) - et al.
An iterative algorithm for least squares problem in quaternionic quantum theory
Comput. Phys. Commun.
(2008) - et al.
Gaussian ensemble of tridiagonal symmetric random matrices
Phys. Lett. A
(1997) Analytic solution of the wave equation for an electron in the field of a molecule with an electric dipole moment
Ann. Phys.
(2008)Quaternionic Quantum Mechanics and Quantum Fields
(1994)Quaternionic quantum field theory
Commun. Math. Phys.
(1986)
Neutron interferometrix search for quaternions in quantum mechanics
Phys. Rev. A
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