论文信息 - The general common Hermitian nonnegative-definite solution to the matrix equations AXA * = BB * and CXC * = DD * with applications in statistics

The general common Hermitian nonnegative-definite solution to the matrix equations AXA * = BB * and CXC * = DD * with applications in statistics

We deduce a necessary and sufficient condition for the matrix equations AXA* = BB* and CXC* = DD* to have a common Hermitian nonnegative-definite solution, and a representation of the general common Hermitian nonnegative-definite solution to these two equations when they have such common solutions. Thereby, we solve a statistical problem which is concerned in testing linear hypotheses about regression coefficients in the multivariate linear model. This paper is a revision of Young et al. (J. Multivariate Anal. 68 (1999) 165) whose mistake was pointed out in (Linear Algebra Appl. 321 (2000) 123).

Xian Zhang | Xian Zhang

[1] A. R. Rao,et al. Linear algebra , 1961 .

[2] Xian Zhang,et al. The rank-constrained Hermitian nonnegative-definite and positive-definite solutions to the matrix equation AXA ∗ = B , 2003 .

[3] J. K. Baksalary,et al. Nonnegative definite and positive definite solutions to the matrix equation AXA * = B , 1984 .

[4] K. S. Banerjee. Generalized Inverse of Matrices and Its Applications , 1973 .

[5] Dean M. Young,et al. Independence Distribution Preserving Covariance Structures for the Multivariate Linear Model , 1999 .

[6] Jürgen Groß,et al. Nonnegative-definite and positive-definite solutions to the matrix equationAXA∗=B – revisited , 2000 .

[7] Sujit Kumar Mitra,et al. Hermitian and Nonnegative Definite Solutions of Linear Matrix Equations , 1976 .

[8] G. Marsaglia,et al. When Does Rank(A+B)=Rank(A)+Rank(B)? , 1972, Canadian Mathematical Bulletin.

[9] Peter Lancaster,et al. Linear matrix equations from an inverse problem of vibration theory , 1996 .

[10] A. Zinober. Matrices: Methods and Applications , 1992 .