Algebraic algorithms for least squares problem in quaternionic quantum theory

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Abstract

Quaternionic least squares (QLS) problem is one method of solving overdetermined sets of quaternion linear equations AXB that is appropriate when there is error in the matrix B. In this paper, by means of complex representation of a quaternion matrix, we introduce a concept of norm of quaternion matrices, discuss singular values and generalized inverses of a quaternion matrix, study the QLS problem and derive two algebraic methods for finding solutions of the QLS problem in quaternionic quantum theory.

Introduction

Applications of quaternions and quaternion matrices have been getting important and extensive in quaternionic quantum mechanics and field theory [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12]. In the study of theory and numerical computations of quaternionic quantum theory, in order to well understand the perturbation theory [5], experimental proposals [7], [8], [9] and theoretical discussions [10], [11], [12] underlying the quaternionic formulations of the Schrödinger equation and so on, one often meets problems of approximate solutions of quaternion problems, such as approximate solution of quaternion linear equations AXB that is appropriate when there is error in the matrix B, i.e. quaternionic least squares (QLS) problem. The main difficulty in obtaining quaternionic approximate solutions of a physical problem is due to the fact of the non-commutation of quaternion in general, and the standard mathematical methods of resolution break down. It is known that the complex least squares (LS) problem has been developed as a global fitting technique especially in physics for solving approximate solutions of complex linear equations AXB if errors occur in the matrix B. But the QLS problem has not been settled now.

In [13], [14], [15], we studied the problems of eigenvalues and eigenvectors of quaternion matrices, quaternionic linear equations and diagonalization of quaternion matrices in quaternionic quantum theory by means of complex representations of quaternion matrices, derived new algebraic methods for the quaternionic matrix problems in quaternionic quantum mechanics and field theory. In this paper, we introduce a concept of norm of a quaternion matrix by means of complex representation of a quaternion matrix, discuss singular values and generalized inverses of a quaternion matrix, study the QLS problem and derive two algebraic methods for finding solutions of the QLS problem in quaternionic quantum mechanics and field theory.

Let R denote the real number field, C the complex number field, Q=RRiRjRk the quaternion field, where i2=j2=k2=1, ij=ji=k. For any quaternion x=x1+x2i+x3j+x4k where xiR, the conjugate of quaternion x is x¯=x1x2ix3jx4k. For any quaternion matrix A, AT, A¯ and AH denote the transpose, conjugate and conjugate transpose of A over quaternion field, respectively. Fm×n denotes the set of m×n matrices on a field F. For any quaternion matrix A=B1+B2i+B3j+B4kQm×n, BlRm×n, l=1,2,3,4, A can be uniquely written as A=(B1+B2i)+(B3+B4i)j=A1+A2j, A1,A2Cm×n. It is easy to verify that for any ACm×n, we have Aj=jA¯, and jA¯j=A. For any AQn×n, A is unitary if AHA=In.

Let us first recall the complex least squares question. In the complex least squares (LS) question we are given an m×n data matrix A, an m×p matrix of observation B, and are asked to find a complex matrix X such thatAXB2=min. Here 2 denotes Euclidean norm. It is well known that the solution to the LS question is unique if rank(A)=n. However, regardless of the rank of A there is always a unique minimal 2 norm solution to the LS question given byXLS=A+B, where A+ denotes the Moore–Penrose pseudo-inverse of complex matrix A.

Section snippets

Norms of quaternion matrices

For any quaternion matrix A=A1+A2jQm×n, in Ref. [15], the author defined a complex representationAf=[A1A2A¯2A¯1]C2m×2n, the complex matrix Af was called complex representation of A.

Let A,BQm×n, CQn×s, aR. From [15] we have following results.(A+B)f=Af+Bf,(aA)f=aAf,(AC)f=AfCf,(AH)f=(Af)H,Af¯=Qm−1AfQn, where Qt=[0ItIt0] is a unitary matrix, It is t×t identity matrix.

For AQm×m, by (2.3) we easily know that the quaternion matrix A is nonsingular if and only if Af is nonsingular and the

Singular values of a quaternion matrix

For any quaternion matrix AQm×n, by [16] we know that the eigenvalues of quaternion matrix AHA are all nonnegative real numbers. The nonnegative square roots of the n eigenvalues of quaternion matrix AHA are called the singular values of quaternion matrix A. [17] gave the following singular value decomposition (SVD) theorem of quaternion matrices by construction.

Proposition 3.1 SVD

(See [17].) Let AQm×n, and r=rank(A). Then there exist unitary quaternion matrices UQm×m and VQn×n such thatUAV=[Σ000]Qm×n, where

Generalized inverses of a quaternion matrix

In order to study the QLS problem, this section introduces generalized inverses of a quaternion matrix, and discuss relations of generalized inverses between a quaternion matrix and its complex representation matrix.

For a quaternion matrix AQm×n, a generalized inverse of A is a quaternion matrix X with following Penrose conditions(1)AXA=A,(3)(AX)H=AX,(2)XAX=X,(4)(XA)H=XA.

Definition 4.1

For AQm×n, XQn×m is said to be a (i,j,) generalized inverse of A if X satisfies Penrose conditions (i),(j), in Eq. (4.1)

Generalized inverse method

In the study of theory and practical numerical computations of applied quaternion disciplines, one meets the following problems of approximate solution of quaternion linear equations AX=B, in which AQm×n, BQm×s. In some cases, the equation has no precision solutions, and one have to find approximate solutions X of quaternion linear equations AX=B, i.e.AXB2=min. Above problem is called to be the quaternionic least squares problem (QLS problem, for short), the solutions of (5.1) are said to

Complex formula method

In this section, we study the quaternion least squares (QLS) problem by means of complex representations of quaternion matrices, and give another way of finding a solution of the QLS problem.

Theorem 6.1

The QLS problem (5.1) has a solution XQn×p if and only if complex LS question (5.2) has a solution YC2n×2p, in which case, if Y is a solution to complex LS question (5.2), then following matrix is a solution to QLS problem (5.1):X=14(In,jIn)(Y+Qn−1Y¯Qp)(IpjIp). Moreover, if rank(A)=n, then the QLS problem

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This paper is partly supported by the National Natural Science Foundation of China (10671086) and Shandong Natural Science Foundation of China (Y2005A12).

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