## Analysis of transformations of linear random-effects models

Assume that a linear random-effects model (LRM) $\by = \bX \bbe + \bve = \bX\bbe+ \bve$ with $\bbe = \bA \bal + \bga$ is transformed as $\bT\by = \bT\bX\bbe + \bT\bve = \bT\bX\bA \bal + \bT\bX\bga + \bT\bve$ by pre-multiplying a given matrix $\bT$. Estimations/predictions of the unknown parameters under the two models are not … Read more

## A new algebraic analysis to linear mixed models

This article presents a new investigation to the linear mixed model $\by = \bX \bbe + \bZ\bga + \bve$ with fixed effect $\bX\bbe$ and random effect $\bZ\bga$ under a general assumption via some novel algebraic tools in matrix theory, and reveals a variety of deep and profound properties hidden behind the linear mixed model. We … Read more

## Solutions of a constrained Hermitian matrix-valued function optimization problem with applications

Let $f(X) =\left( XC + D\right)M\left(XC + D \right)^{*} – G$ be a given nonlinear Hermitian matrix-valued function with $M = M^*$ and $G = G^*$, and assume that the variable matrix $X$ satisfies the consistent linear matrix equation $XA = B$. This paper shows how to characterize the semi-definiteness of $f(X)$ subject to all … Read more

## Analytical formulas for calculating the extremal ranks and inertias of + BXB^{*}$when$ is a fixed-rank Hermitian matrix

The rank of a matrix and the inertia of a square matrix are two of the most generic concepts in matrix theory for describing the dimension of the row/column vector space and the sign distribution of the eigenvalues of the matrix. Matrix rank and inertia optimization problems are a class of discontinuous optimization problems, in … Read more

## Analytical formulas for calculating extremal ranks and inertias of quadratic matrix-valued functions

group of analytical formulas formulas for calculating the global maximal and minimal ranks and inertias of the quadratic matrix-valued function $$\phi(X) = \left(\, AXB + C\,\right)\!M\!\left(\, AXB + C \right)^{*} + D$$ are established and their consequences are presented, where $A$, $B$, $C$ and $D$ are given complex matrices with $A$ and $C$ … Read more

## Analytical formulas for calculating the extremal ranks of the matrix-valued function + BXC$when the rank of$ is fixed

Analytical formulas are established for calculating the maximal and minimal ranks of the matrix-valued function $A + BXC$ when the rank of $X$ is fixed. Some consequences are also given. Article Download View Analytical formulas for calculating the extremal ranks of the matrix-valued function + BXC$when the rank of$ is fixed

## On global optimizations of the rank and inertia of the matrix function $A_1- B_1XB^*_1$ subject to a pair of matrix equations $[\,B_2XB^*_2, \, B_3XB^*_3 \,] = [\,A_2, \, A_3\,]$

For a given linear matrix function $A_1 – B_1XB^*_1$, where $X$ is a variable Hermitian matrix, this paper derives a group of closed-form formulas for calculating the global maximum and minimum ranks and inertias of the matrix function subject to a pair of consistent matrix equations $B_2XB^*_2 = A_2$ and $B_3XB_3^* = A_3$. As applications, … Read more

## Max-min optimizations on the rank and inertia of a linear Hermitian matrix expression subject to range, rank and definiteness restrictions

The inertia of a Hermitian matrix is defined to be a triplet composed by the numbers of the positive, negative and zero eigenvalues of the matrix counted with multiplicities, respectively. In this paper, we give various closed-form formulas for the maximal and minimal values for the rank and inertia of the Hermitian expression \$A + … Read more