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Yongge Tian

Publications and source records attributed to Yongge Tian.

15 recordsLinked to original sources
arXiv2019

Reverse order laws for generalized inverses of products of two or three matrices with applications

One of the fundamental research problems in the theory of generalized inverses of matrices is to establish reverse order laws for generalized inverses of matrix products. Under the assumption that $A$, $B$, and $C$ are three nonsingular matrices of the same size, the products $AB$ and $ABC$ are nonsingular as well, and the inverses of $AB$ and $ABC$ admit the reverse order laws $(AB)^{-1} = B^{-1} A^{-1}$ and $(ABC)^{-1} = C^{-1}B^{-1}A^{-1}$, respectively. If some or all of $A$, $B$, and $C$ are singular, two extensions of the above reverse order laws to generalized inverses can be written as $(AB)^{(i,\ldots,j)} = B^{(i_2,\ldots,j_2)} A^{(i_1,\ldots,j_1)}$ and $(ABC)^{(i,\ldots,j)} = C^{(i_3,\ldots,j_3)} B^{(i_2,\ldots,j_2)}A^{(i_1,\ldots,j_1)}$, or other mixed reverse order laws. These equalities do not necessarily hold for different choices of generalized inverses of the matrices. Thus it is a tremendous work to classify and derive necessary and sufficient conditions for the reverse order law to hold because there are all 15 types of $\{i,\ldots, j\}$-generalized inverse for a given matrix according to combinatoric choices of the four Penrose equations. In this paper, we first establish several decades of mixed reverse order laws for $\{1\}$- and $\{1,2\}$-generalized inverses of $AB$ and $ABC$. We then give a classified investigation to a special family of reverse order laws $(ABC)^{(i,\ldots,j)} = C^{-1}B^{(k,\ldots,l)}A^{-1}$ for the eight commonly-used types of generalized inverses using definitions, formulas for ranges and ranks of matrices, as well as conventional operations of matrices. Furthermore, the special cases $(ABA^{-1})^{(i,\ldots,j)} = AB^{(k,\ldots,l)}A^{-1}$ are addressed and some applications are presented.

math.GM↗
arXiv2019

Twists of two or multiple idempotent matrices

In this article, we revisit some block matrix construction methods and use them to derive various general expansion formulas for calculating the ranks of matrix expressions. As applications, we derive a variety of interesting rank equalities for matrix expressions composed by idempotent matrices, and present their applications in the characterization of some matrix equalities for generalized inverses of partitioned matrices.

math.GM↗
arXiv2013

Solutions of the matrix inequalities $BXB^* <=^- A$ in the minus partial ordering and $BXB^* <=^L A$ in the Löwner partial ordering

Two matrices $A$ and $B$ of the same size are said to satisfy the minus partial ordering, denoted by $B\leqslant^{-}A$, iff the rank subtractivity equality ${\rm rank}(\, A - B\,) = {\rm rank}(A) -{\rm rank}(B)$ holds; two complex Hermitian matrices $A$ and $B$ of the same size are said to satisfy the Löwner partial ordering, denoted by $B\leqslant^{\rm L} A$, iff the difference $A - B$ is nonnegative definite. In this note, we establish general solution of the inequality $BXB^{*} \leqslant^{-}A$ induced from the minus partial ordering, and general solution of the inequality $BXB^{*} \leqslant^{\rm L} A$ induced from the Löwner partial ordering, respectively, where $(\cdot)^{*}$ denotes the conjugate transpose of a complex matrix. As consequences, we give closed-form expressions for the shorted matrices of $A$ relative to the range of $B$ in the minus and Löwner partial orderings, respectively, and show that these two types of shorted matrices in fact are the same.

math.RA↗
arXiv2013

Equalities and inequalities for Hermitian solutions and Hermitian definite solutions of the two matrix equations $AX = B$ and $AXA^* = B$

This paper studies algebraic properties of Hermitian solutions and Hermitian definite solutions of the two types of matrix equation $AX = B$ and $AXA^* = B$. We first establish a variety of rank and inertia formulas for calculating the maximal and minimal ranks and inertias of Hermitian solutions and Hermitian definite solutions of the matrix equations $AX = B$ and $AXA^* = B$, and then use them to characterize many qualities and inequalities for Hermitian solutions and Hermitian definite solutions of the two matrix equations and their variations.

math.RA↗
arXiv2013

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

One of the simplest matrix-valued function with a single variable matrix $X$ is given by $A + BXC$. In this this note, analytical formulas are established for calculating the maximal and minimal ranks of $A + BXC$ when the rank of the variable matrix $X$ is fixed by using a simultaneous decomposition of $A$, $B$ and $C$ and some preliminary results. Some applications of the formulas in completing partially-specified block matrix with the maximal and minimal ranks are also given.

math.OC↗
arXiv2013

Formulas for calculating the extremal ranks and inertias of a matrix-valued function subject to matrix equation restrictions

Matrix rank and inertia optimization problems are a class of discontinuous optimization problems in which the decision variables are matrices running over certain matrix sets, while the ranks and inertias of the variable matrices are taken as integer-valued objective functions. In this paper, we establish a group of explicit formulas for calculating the maximal and minimal values of the rank and inertia objective functions of the Hermitian matrix expression $A_1 - B_1XB_1^{*}$ subject to the common Hermitian solution of a pair of consistent matrix equations $B_2XB^{*}_2 = A_2$ and $B_3XB_3^{*} = A_3$, and Hermitian solution of the consistent matrix equation $B_4X= A_4$, respectively. Many consequences are obtained, in particular, necessary and sufficient conditions are established for the triple matrix equations $B_1XB^{*}_1 =A_1$, $B_2XB^{*}_2 = A_2$ and $B_3XB^{*}_3 = A_3$ to have a common Hermitian solution, as necessary and sufficient conditions for the two matrix equations $B_1XB^{*}_1 =A_1$ and $B_4X = A_4$ to have a common Hermitian solution.

math.OC↗
arXiv2013

Closed-form formulas for calculating the extremal ranks and inertias of a quadratic matrix-valued function and their applications

This paper presents a group of analytical formulas for calculating the global maximal and minimal ranks and inertias of the quadratic matrix-valued function $ϕ(X) = (\, AXB + C\,)M(\, AXB + C)^{*} + D$ and use them to derive necessary and sufficient conditions for the two types of multiple quadratic matrix-valued function {align*} (\, \sum_{i = 1}^{k}A_iX_iB_i + C \,)M(\,\sum_{i = 1}^{k}A_iX_iB_i + C \,)^{*} +D, \ \ \ \sum_{i = 1}^{k}(\,A_iX_iB_i + C_i\,)M_i(\,A_iX_iB_i + C_i \,)^{*} +D {align*} to be semi-definite, respectively, where $A_i,\ B_i,\ C_i,\ C,\ D,\ M_i$ and $M$ are given matrices with $M_i$, $M$ and $D$ Hermitian, $i =1,..., k$. Löwner partial ordering optimizations of the two matrix-valued functions are studied and their solutions are characterized.

math.OC↗
arXiv2013

Analytical solutions to some optimization problems on ranks and inertias of matrix-valued functions subject to linear matrix inequalities

Matrix rank and inertia optimization problems are a class of discontinuous optimization problems, in which the decision variables are matrices running over certain feasible matrix sets, while the ranks and inertias of the variable matrices are taken as integer-valued objective functions. In this paper, we establish a group of explicit formulas for calculating the maximal and minimal values of the rank- and inertia-objective functions of the Hermitian matrix expression $A_1 - B_1XB_1^{*}$ subject to the linear matrix inequality $B_2XB_2^{*} \succcurlyeq A_2$ $(B_2XB_2^{*} \preccurlyeq A_2)$ in the Löwner partial ordering, and give applications of these formulas in characterizing behaviors of some constrained matrix-valued functions.

math.OC↗
arXiv2000

Matrix Representations of Octonions and Their Applications

As is well-known, the real quaternion division algebra $ {\cal H}$ is algebraically isomorphic to a 4-by-4 real matrix algebra. But the real division octonion algebra ${\cal O}$ can not be algebraically isomorphic to any matrix algebras over the real number field ${\cal R}$, because ${\cal O}$ is a non-associative algebra over ${\cal R}$. However since ${\cal O}$ is an extension of ${\cal H}$ by the Cayley-Dickson process and is also finite-dimensional, some pseudo real matrix representations of octonions can still be introduced through real matrix representations of quaternions. In this paper we give a complete investigation to real matrix representations of octonions, and consider their various applications to octonions as well as matrices of octonions.

math.RA↗
arXiv2000

Matrix Theory over the Complex Quaternion Algebra

We present in this paper some fundamental tools for developing matrix analysis over the complex quaternion algebra. As applications, we consider generalized inverses, eigenvalues and eigenvectors, similarity, determinants of complex quaternion matrices, and so on.

math.RA↗
arXiv2000

Rank Equalities Related to Generalized Inverses of Matrices and Their Applications

This paper is divided into two parts. In the first part, we develop a general method for expressing ranks of matrix expressions that involve Moore-Penrose inverses, group inverses, Drazin inverses, as well as weighted Moore-Penrose inverses of matrices. Through this method we establish a variety of valuable rank equalities related to generalized inverses of matrices mentioned above. Using them, we characterize many matrix equalities in the theory of generalized inverses of matrices and their applications. In the second part, we consider maximal and minimal possible ranks of matrix expressions that involve variant matrices, the fundamental work is concerning extreme ranks of the two linear matrix expressions $A - BXC$ and $A - B_1X_1C_1 - B_2X_2C_2$. As applications, we present a wide range of their consequences and applications in matrix theory.

math.RA↗