Nonnegativity of the Moore-Penrose inverse of a perturbation of the form $A-XG{Y}^T$ is considered when $A^{\mathrm{†}}\ge 0$. Using a generalized version of the Sherman-Morrison-Woodbury formula, conditions for ${(A-XG{Y}^T)}^{\mathrm{†}}$ to be nonnegative are derived. Applications of the results are presented briefly. Iterative versions of the results are also studied.

Multisender authentication codes allow a group of senders to construct an authenticated message for a receiver such that the receiver can verify the authenticity of the received message. In this paper, we construct one multisender authentication code from polynomials over finite fields. Some parameters and the probabilities of deceptions of this code are also computed.

We analyze the best approximation $AN$ (in the Frobenius sense) to the identity matrix in an arbitrary matrix subspace $AS$ ($A\in {ℝ}^{n\times n}$ nonsingular, $S$ being any fixed subspace of ${ℝ}^{n\times n}$). Some new geometrical and spectral properties of the orthogonal projection $AN$ are derived. In particular, new inequalities for the trace and for the eigenvalues of matrix $AN$ are presented for the special case that $AN$ is symmetric and positive definite.

By using Sherman-Morrison-Woodbury formula, we introduce a preconditioner based on parameterized splitting idea for generalized saddle point problems which may be singular and nonsymmetric. By analyzing the eigenvalues of the preconditioned matrix, we find that when α is big enough, it has an eigenvalue at 1 with multiplicity at least $n$, and the remaining eigenvalues are all located in a unit circle centered at 1. Particularly, when the preconditioner is used in general saddle point problems, it guarantees eigenvalue at 1 with the same multiplicity, and the remaining eigenvalues will tend to 1 as the parameter $\alpha \to 0$. Consequently, this can lead to a good convergence when some GMRES iterative methods are used in Krylov subspace. Numerical results of Stokes problems and Oseen problems are presented to illustrate the behavior of the preconditioner.

We consider the extremal inertias and ranks of the matrix expressions $f(X,Y)=A_{\mathrm{3}}-B_{\mathrm{3}}X-(B_{\mathrm{3}}X{)}^{\mathrm{*}}-C_{\mathrm{3}}Y{D}_{\mathrm{3}}-(C_{\mathrm{3}}Y{D}_{\mathrm{3}}{)}^{\mathrm{*}}$, where $A_{\mathrm{3}}=A_{\mathrm{3}}^{\mathrm{*}},$ $B_{\mathrm{3}},$ $C_{\mathrm{3}}$, and $D_{\mathrm{3}}$ are known matrices and $Y$ and $X$ are the solutions to the matrix equations $A_{\mathrm{1}}Y=C_{\mathrm{1}}$, $Y{B}_{\mathrm{1}}=D_{\mathrm{1}}$, and $A_{\mathrm{2}}X=C_{\mathrm{2}}$, respectively. As applications, we present necessary and sufficient condition for the previous matrix function $f(X,Y)$ to be positive (negative), non-negative (positive) definite or nonsingular. We also characterize the relations between the Hermitian part of the solutions of the above-mentioned matrix equations. Furthermore, we establish necessary and sufficient conditions for the solvability of the system of matrix equations $A_{\mathrm{1}}Y=C_{\mathrm{1}}$, $Y{B}_{\mathrm{1}}=D_{\mathrm{1}}$, $A_{\mathrm{2}}X=C_{\mathrm{2}}$, and $B_{\mathrm{3}}X+(B_{\mathrm{3}}X{)}^{\mathrm{*}}+C_{\mathrm{3}}Y{D}_{\mathrm{3}}+(C_{\mathrm{3}}Y{D}_{\mathrm{3}}{)}^{\mathrm{*}}=A_{\mathrm{3}}$, and give an expression of the general solution to the above-mentioned system when it is solvable.

Left and right inverse eigenpairs problem for $\kappa$-hermitian matrices and its optimal approximate problem are considered. Based on the special properties of $\kappa$-hermitian matrices, the equivalent problem is obtained. Combining a new inner product of matrices, the necessary and sufficient conditions for the solvability of the problem and its general solutions are derived. Furthermore, the optimal approximate solution and a calculation procedure to obtain the optimal approximate solution are provided.

This paper considers a completion problem of a nonsingular $2\times 2$ block matrix over the real quaternion algebra $ℍ$: Let $m_1,m_2,n_1,n_2$ be nonnegative integers, $m_1+m_2=n_1+n_2=n>0$, and $A_{12}\in {ℍ}^{m_1\times n_2},\mathrm{ }{A}_{21}\in {ℍ}^{m_2\times n_1},\mathrm{ }{A}_{22}\in {ℍ}^{m_2\times n_2},\mathrm{ }{B}_{11}\in {ℍ}^{n_1\times m_1}$ be given. We determine necessary and sufficient conditions so that there exists a variant block entry matrix $A_{11}\in {ℍ}^{m_1\times n_1}$ such that is nonsingular, and $B_{11}$ is the upper left block of a partitioning of $A^{-1}$. The general expression for $A_{11}$ is also obtained. Finally, a numerical example is presented to verify the theoretical findings.

The fuzzy matrix equations $\tilde{A}\otimes \tilde{X}\otimes \tilde{B}=\tilde{C}$ in which $\tilde{A}$, $\tilde{B}$, and $\tilde{C}$ are $m\times m$, $n\times n$, and $m\times n$ nonnegative LR fuzzy numbers matrices, respectively, are investigated. The fuzzy matrix systems is extended into three crisp systems of linear matrix equations according to arithmetic operations of LR fuzzy numbers. Based on pseudoinverse of matrix, the fuzzy approximate solution of original fuzzy systems is obtained by solving the crisp linear matrix systems. In addition, the existence condition of nonnegative fuzzy solution is discussed. Two examples are calculated to illustrate the proposed method.

We establish the formulas of the maximal and minimal ranks of the quaternion Hermitian matrix expression $C_4-A_{4}X{A}_4^{\ast }$ where $X$ is a Hermitian solution to quaternion matrix equations $A_{1}X=C_1$, $X{B}_1=C_2$, and $A_{3}X{A}_3^{*}=C_3$. As applications, we give a new necessary and sufficient condition for the existence of Hermitian solution to the system of matrix equations $A_{1}X=C_1$, $X{B}_1=C_2$, $A_{3}X{A}_3^{*}=C_3$, and $A_{4}X{A}_4^{*}=C_4$, which was investigated by Wang and Wu, 2010, by rank equalities. In addition, extremal ranks of the generalized Hermitian Schur complement $C_4-A_4{A}_3^{~}{A}_4^{\ast }$ with respect to a Hermitian g-inverse $A_3^{~}$ of $A_3$, which is a common solution to quaternion matrix equations $A_{1}X=C_1$ and $X{B}_1=C_2$, are also considered.

Erratum to “Ranks of a Constrained Hermitian Matrix Expression with Applications” dx.doi.org/10.1155/2013/826397

We propose an iterative algorithm for solving the reflexive solution of the quaternion matrix equation $AXB+C{X}^{H}D=F$. When the matrix equation is consistent over reflexive matrix $X$, a reflexive solution can be obtained within finite iteration steps in the absence of roundoff errors. By the proposed iterative algorithm, the least Frobenius norm reflexive solution of the matrix equation can be derived when an appropriate initial iterative matrix is chosen. Furthermore, the optimal approximate reflexive solution to a given reflexive matrix $X_0$ can be derived by finding the least Frobenius norm reflexive solution of a new corresponding quaternion matrix equation. Finally, two numerical examples are given to illustrate the efficiency of the proposed methods.

We solve optimization problems on the ranks and inertias of the quadratic Hermitian matrix function $Q-XP{X}^{*}$ subject to a consistent system of matrix equations $AX=C$ and $XB=D$. As applications, we derive necessary and sufficient conditions for the solvability to the systems of matrix equations and matrix inequalities $AX=C,XB=D$, and in the Löwner partial ordering to be feasible, respectively. The findings of this paper widely extend the known results in the literature.