EPr Solution to a System of Matrix Equations

Affiliation(s)

School of Mathematics and Science, Shijiazhuang University of Economics, Shijiazhuang, China.

School of Mathematics and Science, Shijiazhuang University of Economics, Shijiazhuang, China.

ABSTRACT

A square complex matrix is called if it can be written in the form with being fixed unitary and being arbitrary matrix in . We give necessary and sufficient conditions for the existence of the solution to the system of complex matrix equation and present an expression of the solution to the system when the solvability conditions are satisfied. In addition, the solution to an optimal approximation problem is obtained. Furthermore, the least square solution with least norm to this system mentioned above is considered. The representation of such solution is also derived.

KEYWORDS

EP Matrix; Matrix Equation; Moore-Penrose Inverse; Approximation Problem; Least Squares Solution

EP Matrix; Matrix Equation; Moore-Penrose Inverse; Approximation Problem; Least Squares Solution

Cite this paper

C. Dong, Y. Zhang and J. Song, "EPr Solution to a System of Matrix Equations,"*Advances in Linear Algebra & Matrix Theory*, Vol. 3 No. 4, 2013, pp. 50-54. doi: 10.4236/alamt.2013.34010.

C. Dong, Y. Zhang and J. Song, "EPr Solution to a System of Matrix Equations,"

References

[1] H. Schwerdtfeger, “Introduction to Linear Algebra and the Theory of Matrices,” P. Noordhoff, Groningen, 1950.

[2] M. H. Pearl, “On normal and matrices,” Michigan Mathematical Journal, Vol. 6, No. 1, 1959, pp. 1-5. http://dx.doi.org/10.1307/mmj/1028998132

[3] C. R. Rao and S. K. Mitra, “Generalized Inverse of Matrices and Its Applications,” Wiley, New York, 1971.

[4] O. M. Baksalary and G. Trenkler, “Characterizations of EP, normal, and Hermitian matrices,” Linear Multilinear Algebra, Vol. 56, 2008, pp. 299-304. http://dx.doi.org/10.1080/03081080600872616

[5] Y. Tian and H. X. Wang, “Characterizations of Matrices and Weighted-EP Matrices,” Linear Algebra Applications, Vol. 434, No. 5, 2011, pp. 1295-1318. http://dx.doi.org/10.1016/j.laa.2010.11.014

[6] K.-W. E. Chu, “Singular Symmetric Solutions of Linear Matrix Equations by Matrix Decompositions,” Linear Algebra Applications, Vol. 119, 1989, pp. 35-50. http://dx.doi.org/10.1016/0024-3795(89)90067-0

[7] R. D. Hill, R. G. Bates and S. R. Waters, “On Centrohermitian Matrices,” SIAM Journal on Matrix Analysis and Applications, Vol. 11, No. 1, 1990, pp. 128-133. http://dx.doi.org/10.1137/0611009

[8] Z. Z. Zhang, X. Y. Hu and L. Zhang, “On the Hermitian-Generalized Hamiltonian Solutions of Linear Mattrix Equations,” SIAM Journal on Matrix Analysis and Applications, Vol. 27, No. 1, 2005, pp. 294-303. http://dx.doi.org/10.1137/S0895479801396725

[9] A. Dajic and J. J. Koliha, “Equations and in Rings and Rings with Involution with Applications to Hilbert Space Operators,” Linear Algebra Applications, Vol. 429, No. 7, 2008, pp. 1779-1809. http://dx.doi.org/10.1016/j.laa.2008.05.012

[10] C. G. Khatri and S. K. Mitra, “Hermitian and Nonnegative Definite Solutions of Linear Matrix Equations,” SIAM Journal on Matrix Analysis and Applications, Vol. 31, No. 4, 1976, pp. 579-585. http://dx.doi.org/10.1137/0131050

[11] F. J. H. Don, “On the Symmetric Solutions of a Linear Matrix Equation,” Linear Algebra Applications, Vol. 93, 1987, pp. 1-7. http://dx.doi.org/10.1016/S0024-3795(87)90308-9

[12] H. X. Chang, Q. W. Wang and G. J. Song, “(R,S)-Conjugate Solution to a Pair of Linear Matrix Equations,” Applied Mathematics and Computation, Vol. 217, 2010, pp. 73-82. http://dx.doi.org/10.1016/j.amc.2010.04.053

[13] E. W. Cheney, “Introduction to Approximation Theory,” McGraw-Hill Book Co., 1966.

[1] H. Schwerdtfeger, “Introduction to Linear Algebra and the Theory of Matrices,” P. Noordhoff, Groningen, 1950.

[2] M. H. Pearl, “On normal and matrices,” Michigan Mathematical Journal, Vol. 6, No. 1, 1959, pp. 1-5. http://dx.doi.org/10.1307/mmj/1028998132

[3] C. R. Rao and S. K. Mitra, “Generalized Inverse of Matrices and Its Applications,” Wiley, New York, 1971.

[4] O. M. Baksalary and G. Trenkler, “Characterizations of EP, normal, and Hermitian matrices,” Linear Multilinear Algebra, Vol. 56, 2008, pp. 299-304. http://dx.doi.org/10.1080/03081080600872616

[5] Y. Tian and H. X. Wang, “Characterizations of Matrices and Weighted-EP Matrices,” Linear Algebra Applications, Vol. 434, No. 5, 2011, pp. 1295-1318. http://dx.doi.org/10.1016/j.laa.2010.11.014

[6] K.-W. E. Chu, “Singular Symmetric Solutions of Linear Matrix Equations by Matrix Decompositions,” Linear Algebra Applications, Vol. 119, 1989, pp. 35-50. http://dx.doi.org/10.1016/0024-3795(89)90067-0

[7] R. D. Hill, R. G. Bates and S. R. Waters, “On Centrohermitian Matrices,” SIAM Journal on Matrix Analysis and Applications, Vol. 11, No. 1, 1990, pp. 128-133. http://dx.doi.org/10.1137/0611009

[8] Z. Z. Zhang, X. Y. Hu and L. Zhang, “On the Hermitian-Generalized Hamiltonian Solutions of Linear Mattrix Equations,” SIAM Journal on Matrix Analysis and Applications, Vol. 27, No. 1, 2005, pp. 294-303. http://dx.doi.org/10.1137/S0895479801396725

[9] A. Dajic and J. J. Koliha, “Equations and in Rings and Rings with Involution with Applications to Hilbert Space Operators,” Linear Algebra Applications, Vol. 429, No. 7, 2008, pp. 1779-1809. http://dx.doi.org/10.1016/j.laa.2008.05.012

[10] C. G. Khatri and S. K. Mitra, “Hermitian and Nonnegative Definite Solutions of Linear Matrix Equations,” SIAM Journal on Matrix Analysis and Applications, Vol. 31, No. 4, 1976, pp. 579-585. http://dx.doi.org/10.1137/0131050

[11] F. J. H. Don, “On the Symmetric Solutions of a Linear Matrix Equation,” Linear Algebra Applications, Vol. 93, 1987, pp. 1-7. http://dx.doi.org/10.1016/S0024-3795(87)90308-9

[12] H. X. Chang, Q. W. Wang and G. J. Song, “(R,S)-Conjugate Solution to a Pair of Linear Matrix Equations,” Applied Mathematics and Computation, Vol. 217, 2010, pp. 73-82. http://dx.doi.org/10.1016/j.amc.2010.04.053

[13] E. W. Cheney, “Introduction to Approximation Theory,” McGraw-Hill Book Co., 1966.