The Matrix Theory And Its Application In Statistical Linear Model

As a powerful tool, matrix rank has many applications. Such as, it can workas a tool to discuss the solvability of linear/nonlinear matrix equation, characterizethe Jordan blocks for the canonical form of matrices in pair and determine whether asystem is controllable or not. In this dissertation, we discuss solutions to some matrixsystem, including linear and nonlinear matrix equation. Besides, we also investigate therelationships of the best linear unbiased estimators(BLUEs)/the best linear unbiasedpredictors (BLUPs) under diferent linear mixed models. Except that, we derive someinteresting results about the (1,j)-inverse, j=3,4. Those results further enrich anddevelop the theory of matrix algebra and its applications in statistical field.This dissertation is divided into5Chapters. In Chapter1, the research back-ground as well as the main contributions of this dissertation are introduced. And somebasic material for later use are also presented. In Chapter2, we derive structuredcanonical forms under the specific contragredient transformation for symmetric/skew-symmetric A, B. As an application, those forms allow us to discuss the existence ofsymmetric solutions X to the quadratic matrix equation XAX=B with A, B (skew-)symmetric matrices. As a result, we know there exists a solution X to the equationXAX=B if and only if Lk≤Rk, where Lk, Rkdenote the number of the left orright zero Jordan pairs of size k in (P1AP T, PTBP), respectively. In Chapter3, weconsider a general mixed linear model M without any rank assumptions to the covari-ance matrix and without any restrictions on the correlation between the random efectsvector and the random errors vector. For the general mixed linear models M1andM2, which have diferent covariance matrices, we derive the necessary and sufcientconditions for that the BLUEs and/or BLUPs under M1continue to be the BLUEsand/or BLUPs under the M2. And we also give the necessary and sufcient condi-tions for the equivalence of BLUP under M1and M2. In Chapter4, we consider thebisymmetric/bisymmetric nonnegative definite solution with extremal ranks/inertias toa system of quaternion matrix equations AX=C, XB=D. We derive the extremalranks/inertias of the common bisymmetric/bisymmetric nonnegative definite solutionto the system. The general expressions of the bisymmetric/bisymmetric nonnegative definite solution with extremal ranks/inertias to the system mentioned above are alsopresented. In addition, we give a numerical example to illustrate the results of this pa-per. In Chapter5, we consider the relationship between the {A(1,j)} and {P N(1,j)Q},where A(1,j), N(1,j)denote the (1, j)-inverse of matrices A, N, respectively, j=3,4.We establish some necessary and sufcient rank conditions for some relationships to behold, say,{A(1,j)}{P N(1,j)Q},{A(1,j)}={P N(1,j)Q}. As applications, we presentsome interesting results about the (1,j)-inverse,j=3,4.