We derive necessary and sufficient conditions for the existence of a Hermitian nonnegative-definite solution to the matrix equation AXB = C. Moreover, we derive a representation of a general Hermitian nonnegative-definite solution. We then apply our solution to two examples, including a comparison of our solution to a proposed solution by Zhang in [1] using an example problem given from [1]. Our solution demonstrates that the proposed general solution from Zhang in [1] is incorrect. We also give a second example in which we derive the general covariance structure so that two matrix quadratic forms are independent.
References
[1]
Zhang, X. (2004) Hermitian Nonnegative-Definite and Positive-Definite Solutions of the Matrix Equation AXB = C. Applied Mathematics E-Notes, 4, 40-47.
[2]
Penrose, R. (1955) A Generalized Inverse for Matrices. Mathematical Proceedings of the Cambridge Philosophical Society, 51, 406-413. https://doi.org/10.1017/s0305004100030401
[3]
Dogaru, O.C. (1985) On the General Solution of the Linear Algebraic Systems. Studia Univeritatea Babes-Bolyai: Mathematica, 30, 29-33.
[4]
Chu, K.W.E. (1987) Singular Value and Generalized Singular Value Decompositions and the Solution of Linear Matrix Equations. Linear Algebra and Its Applications, 88, 83-98. https://doi.org/10.1016/0024-3795(87)90104-2
[5]
Von Rosen, D. (1993) Some Results on Homogeneous Matrix Equations. SIAM Journal on Matrix Analysis and Applications, 14, 137-145. https://doi.org/10.1137/0614013
[6]
Khatri, C.G. (1959) On the Conditions for the Forms of the Type x’ax to Be Distributed Independently or to Obey Wish Art Distribution. Calcutta Statistical Association Bulletin, 8, 162-168. https://doi.org/10.1177/0008068319590404
[7]
Wang, X., Yan, L. and Dai, L. (2013) On Hermitian and Skew-Hermitian Splitting Iteration Methods for the Linear Matrix Equation AXB = C. Computers & Mathematics with Applications, 65, 657-664. https://doi.org/10.1016/j.camwa.2012.11.010
[8]
Cvetkovific-Ilific, D.S. (2006) The Reexive Solutions of the Matrix Equation AXB = C. Computers & Mathematics with Applications, 51, 897-902. https://doi.org/10.1016/j.camwa.2005.11.032
[9]
Wang, Q. and Yang, C. (1998) The Re-Nonnegative Definite Solutions to the Matrix Equation AXB = C. Commentationes Mathematicae Universitalis Carloline, 39, 7-13.
[10]
Cvetkovific-Ilific, D.S. and Dragana, S. (2008) Re-Nnd Solution of the Matrix Equation AXB = C. Journal of the Australian Mathematical Society, 84, 63-72.
[11]
Rao, C.R. and Mitra, S.K. (1971) Generalized Inverse of Matrices and Its Applications. John Wiley & Sons, New York.
[12]
Khatri, C.G. and Mitra, S.K. (1976) Hermitian and Nonnegative Definite Solutions of Linear Matrix Equations. SIAM Journal of Applied Mathematics, 1, 579-585. https://doi.org/10.1137/0131050
[13]
Harville, D.A. (1999) Matrix Algebra from a Statistician’s Perspective. Springer-Verlag, Heidelberg Berlin, New York.
[14]
Albert, A. (1969) Conditions for Positive and Nonnegative Definiteness in Terms of Pseudoinverses. SIAM Journal on Applied Mathematics, 17, 434-440. https://doi.org/10.1137/0117041
[15]
Baksalary, J.K. (1984) Nonnegative Definite and Positive Definite Solutions to the Matrix Equation AXA = B. Linear and Multilinear Algebra, 16, 133-139. https://doi.org/10.1080/03081088408817616
[16]
Grofi, J. (2000) Nonnegative-Definite and Positive-Definite Solutions to the Matrix Equation AXA = B-Revisited. Linear Algebra and Its Applications, 321, 123-129. https://doi.org/10.1016/S0024-3795(00)00033-1
[17]
Mathai, A.M. and Provost, S.B. (1992) Quadratic Forms in Random Variables. Marcel Dekker, New York.
[18]
Gupta, A.K. and Nagar, D.K. (2000) Matrix Variate Distributions. Chapman & Hall/CRC, Boca Raton.
