On Real Matrices to Least-Squares g-Inverse and Minimum Norm g-Inverse of Quaternion Matrices

Author(s)
Huasheng Zhang

ABSTRACT

Through the real representations of quaternion matrices and matrix rank method, we give the expression of the real ma-trices in least-squares g-inverse and minimum norm g-inverse. From these formulas, we derive the extreme ranks of the real matrices. As applications, we establish necessary and sufficient conditions for some special least-squares g-inverse and minimum norm g-inverse.

Through the real representations of quaternion matrices and matrix rank method, we give the expression of the real ma-trices in least-squares g-inverse and minimum norm g-inverse. From these formulas, we derive the extreme ranks of the real matrices. As applications, we establish necessary and sufficient conditions for some special least-squares g-inverse and minimum norm g-inverse.

KEYWORDS

Extreme Rank; g-Inverse; Least-Squares g-Inverse; Minimum Norm g-Inverse; Quaternion Matrix

Extreme Rank; g-Inverse; Least-Squares g-Inverse; Minimum Norm g-Inverse; Quaternion Matrix

Cite this paper

nullH. Zhang, "On Real Matrices to Least-Squares g-Inverse and Minimum Norm g-Inverse of Quaternion Matrices,"*Advances in Linear Algebra & Matrix Theory*, Vol. 1 No. 1, 2011, pp. 1-7. doi: 10.4236/alamt.2011.11001.

nullH. Zhang, "On Real Matrices to Least-Squares g-Inverse and Minimum Norm g-Inverse of Quaternion Matrices,"

References

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[2] A. Ben-Israel, T.N.E. Greville, Generalized in-verses: Theory and Applications, R.E. Krieger Pubilshing Com-pany, New York, 1980.

[3] Haruo Yanai, Some generalized forms a least squares g-inverse, minimum norm g-inverse, and Moore-Penrose inverse matrices, Computational Statistics & Data Analysis. 10 (1990): 251-260.

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[7] S.L. Adler, Quaternionic Quantum Mechanics and Quantum Fields, Oxford University Press, Oxford, 1995.

[8] F. Zhang, Quaternions and matrices of quaternions, Linear Algebra Appl. 251 (1997): 21-57.

[9] C.E. Moxey, S.J. Sangwine, and T.A. Ell, Hypercomplex Correlation Techniques for Vector Images, IEEE Transactions on Signal Processing 51 (7) (2003): 1941-1953.

[10] N. LE Bihan, J. Mars, Singular Value Decomposition of Matrices of Quaternions: A New Tool for Vector-Sensor Signal Processing, Signal Processing 84(7) (2004): 1177-1199.

[11] N. LE Bihan, S.J. Sangwine, Quater-nion Principal Component Analysis of Color images, IEEE International Conference on Image Processing (ICIP), Barce-lona, Spain, September 2003.

[12] N. LE Bihan, S.J. Sangwine, Color Image Decomposition Using Quaternion Singular Value Decomposition, IEEE International conference on Visual In-formation Engineering (VIE), Guildford, UK, July 2003.

[13] S.J. Sangwine, N. LE Bihan, Quaternion singular value decomposition based on bidiagonalization to a real or complex matrix using quaternion Householder transformations, Appl. Math. Comput. 182(1) (2006): 727-738.

[14] Q.W. Wang, H.S. Zhang and S.W. Yu, On solutions to the quaternion matrix equation AXB+CYD=E, Electronic Journal of Linear Algebra. 17 (2008): 343-358.

[15] S.K. Mitra, A pair of si-multaneous linear matrix equations and a matrix programming problem, Linear Algebra and Appl. 131 (1990): 97-123.

[16] Y. Tian, The maximal and minimal ranks of some expressions of generalized inverses of matrices, Southeast Asian Bull. Math. 25 (2002): 745-755.

[1] T.W. Hungerford, Algebra, Spring-Verlag Inc, New York, 1980.

[2] A. Ben-Israel, T.N.E. Greville, Generalized in-verses: Theory and Applications, R.E. Krieger Pubilshing Com-pany, New York, 1980.

[3] Haruo Yanai, Some generalized forms a least squares g-inverse, minimum norm g-inverse, and Moore-Penrose inverse matrices, Computational Statistics & Data Analysis. 10 (1990): 251-260.

[4] Y. Tian, More on maximal and minimal ranks of Schur complements with appli-cations, Applied Mathematics and Computation. 152 (2004): 675-692.

[5] Y. Tian, Schur complements and Ba-nachiewicz-Schur forms, Electronic Journal of Linear Algebra Society. 13 (2005): 405-418.

[6] W Guo, M Wei and M Wang, On least squares g-inverses and minimum norm g-inverses of a bordered matrix, Linear Algebra and Appl. 15 (2006): 627-642.

[7] S.L. Adler, Quaternionic Quantum Mechanics and Quantum Fields, Oxford University Press, Oxford, 1995.

[8] F. Zhang, Quaternions and matrices of quaternions, Linear Algebra Appl. 251 (1997): 21-57.

[9] C.E. Moxey, S.J. Sangwine, and T.A. Ell, Hypercomplex Correlation Techniques for Vector Images, IEEE Transactions on Signal Processing 51 (7) (2003): 1941-1953.

[10] N. LE Bihan, J. Mars, Singular Value Decomposition of Matrices of Quaternions: A New Tool for Vector-Sensor Signal Processing, Signal Processing 84(7) (2004): 1177-1199.

[11] N. LE Bihan, S.J. Sangwine, Quater-nion Principal Component Analysis of Color images, IEEE International Conference on Image Processing (ICIP), Barce-lona, Spain, September 2003.

[12] N. LE Bihan, S.J. Sangwine, Color Image Decomposition Using Quaternion Singular Value Decomposition, IEEE International conference on Visual In-formation Engineering (VIE), Guildford, UK, July 2003.

[13] S.J. Sangwine, N. LE Bihan, Quaternion singular value decomposition based on bidiagonalization to a real or complex matrix using quaternion Householder transformations, Appl. Math. Comput. 182(1) (2006): 727-738.

[14] Q.W. Wang, H.S. Zhang and S.W. Yu, On solutions to the quaternion matrix equation AXB+CYD=E, Electronic Journal of Linear Algebra. 17 (2008): 343-358.

[15] S.K. Mitra, A pair of si-multaneous linear matrix equations and a matrix programming problem, Linear Algebra and Appl. 131 (1990): 97-123.

[16] Y. Tian, The maximal and minimal ranks of some expressions of generalized inverses of matrices, Southeast Asian Bull. Math. 25 (2002): 745-755.