Computation Of Quaternion Matrix Eigenvalue

Quaternion is initiated by W. R. Hamilton, a British mathematician, in 1843. The discovery of quaternion is an importance event in the history of math. Quaternion is applied widely in many domains, such as algebra, geometry, physics, engineering and so on. At present ten years, quaternion is applied more and more in computer science and engineering and gets more and more attention.Matrix computation, the core of science and engineering computation, includes three aspects: solution of linear systems, least squares problems, eigenproblems. Eigenproblems are active problems of computer science and numerical algebra which develop rapidly and have great and wide application in science and engineering.A matrix whose elements are real quaternions is called a real quaternionic matrix (quatemionic matrix for short). The study on quaternionic matrixes has gained great achievements over the past several years [1], [23]-[27], [32], [47], [52], [55]-[57]. Generally speaking, many properties of complex matrixes can be extended to quaternionic matrixes. But quaternionic matrixes also have properties which are different from complex matrixes. Few studies about numerical computation of quaternionic matrixes have been made. The study about the computation of singular eigenvalues of quaternionic matrixes almost hasn't been done because it is too difficult. It has great significance to study eigenproblems of quaternionic matrixes. Let A be a quaternionic matrix,λis called a singular eigenvalue (right eigenvalue) if Ax =λx(Ax = xλ). There are many differences between singular and right eigenvalues of quaternionic matrixes. The study on the right eigenvalues of quaternionic matrixes has gained many satisfied achievements. Bunse-Gerstner [1] gave the QR Factorzation and Schur Decomposition of quaternionic matrixes and got right eigenvalues and right eigenvectors of the matrixes by extending the QR method of complex matrixes to quaternionic matrixes. In this paper, we get the power method about the right eigenvalues of Hermite quaternionic matrixes by extending the power method of real matrixes to Hermite quaternionic matrixes. The computation of the eigenvalues of quaternionic matrixes can translate into the computation of the eigenvalues of its complex representation matrixes. In this paper, we utilize the specific structure of complex representation matrixes to get a method to reduce the quantity of computation of its eigenvalues.There are some new problems about quaternionic matrixes which are not referred in the computation of complex matrixes such as the computation of the singular eigenvalues of quaternionic matrixes. Huang Liping and So Wasin [25] discussed the properties of the singular eigenvalues of quaternionic matrixes and gave the formula of the singular eigenvalues. In this paper, we give a programme programmed by C++ which can compute the singular eigenvalues of every 2×2 quaternionic matrix. In this paper, we discuss the problem of the position estimate of n×n quaternionic matrixes and give the Gerschgorin theorem about singular eigenvalues of quaternionic matrixes. We also give a method to compute the singular eigenvalues of some special quaternionic matrixes in this paper.