The range and existence conditions of the Hermitian positive definite solutions of nonlinear matrix equations Xs+A*X-tA=Q are studied, where A is an n×n non-singular complex matrix and Q is an n×n Hermitian ...The range and existence conditions of the Hermitian positive definite solutions of nonlinear matrix equations Xs+A*X-tA=Q are studied, where A is an n×n non-singular complex matrix and Q is an n×n Hermitian positive definite matrix and parameters s,t>0. Based on the matrix geometry theory, relevant matrix inequality and linear algebra technology, according to the different value ranges of the parameters s,t, the existence intervals of the Hermitian positive definite solution and the necessary conditions for equation solvability are presented, respectively. Comparing the existing correlation results, the proposed upper and lower bounds of the Hermitian positive definite solution are more accurate and applicable.展开更多
By presenting a counterexample, the author of paper （ZHAO Li-feng. J. Math. Res. Exposition, 2007, 27（4）： 949-954） declared that some assertions in papers of LU Yun-xia, ZHANG Shu-qing （J. Math. Res. Exposition,...By presenting a counterexample, the author of paper （ZHAO Li-feng. J. Math. Res. Exposition, 2007, 27（4）： 949-954） declared that some assertions in papers of LU Yun-xia, ZHANG Shu-qing （J. Math. Res. Exposition, 1999, 19（3）： 598-600）, HE Gan-tong （J. Math. Res. Exposition, 2002, 22（1）： 79-82） and YUAN Hui-ping （J. Math. Res. Exposition, 2001, 21（3）： 464-468） are wrong. In this note, we point out that the counterexample is wrong. Further discussion on these assertions and some related results are also given.展开更多
This paper discusses the necessary and sufficient conditions for the existence of Hermite positive definite solutions of the quaternion matrix equation X<sup>m</sup>+ B*XB = C (m > 0) and its iterative ...This paper discusses the necessary and sufficient conditions for the existence of Hermite positive definite solutions of the quaternion matrix equation X<sup>m</sup>+ B*XB = C (m > 0) and its iterative solution method. According to the characteristics of the coefficient matrix, a corresponding algebraic equation system is ingeniously constructed, and by discussing the equation system’s solvability, the matrix equation’s existence interval is obtained. Based on the characteristics of the coefficient matrix, some necessary and sufficient conditions for the existence of Hermitian positive definite solutions of the matrix equation are derived. Then, the upper and lower bounds of the positive actual solutions are estimated by using matrix inequalities. Four iteration formats are constructed according to the given conditions and existence intervals, and their convergence is proven. The selection method for the initial matrix is also provided. Finally, using the complexification operator of quaternion matrices, an equivalent iteration on the complex field is established to solve the equation in the Matlab environment. Two numerical examples are used to test the effectiveness and feasibility of the given method. .展开更多
For the lower bound about the determinant of Hadamard product of A and B, where A is a n × n real positive definite matrix and B is a n × n M-matrix, Jianzhou Liu [SLAM J. Matrix Anal. Appl., 18(2)(1997): 30...For the lower bound about the determinant of Hadamard product of A and B, where A is a n × n real positive definite matrix and B is a n × n M-matrix, Jianzhou Liu [SLAM J. Matrix Anal. Appl., 18(2)(1997): 305-311]obtained the estimated inequality as follows det(A o B)≥a11b11 nⅡk=2(bkk detAk/detAk-1+detBk/detBk-1(k-1Ei=1 aikaki/aii))=Ln(A,B),where Ak is kth order sequential principal sub-matrix of A. We establish an improved lower bound of the form Yn(A,B)=a11baa nⅡk=2(bkk detAk/detAk-1+akk detBk/detBk-1-detAdetBk/detak-1detBk-1)≥Ln(A,B).For more weaker and practical lower bound, Liu given thatdet(A o B)≥(nⅡi=1 bii)detA+(nⅡi=1 aii)detB(nⅡk=2 k-1Ei=1 aikaki/aiiakk)=(L)n(A,B).We further improve it as Yn(A,B)=(nⅡi=1 bii)detA+(nⅡi=1 aii)detB-(detA)(detB)+max1≤k≤n wn(A,B,k)≥(nⅡi=1 bii)detA+(nⅡi=1 aii)detB-(detA)(detB)≥(L)n(A,B).展开更多
In the paper,a necessary and sufficeent condition for generalized diagonal domiance matrices is given.Further, the relations among all generalized positive definite matrices are shown, also,some flaws and mistakes in...In the paper,a necessary and sufficeent condition for generalized diagonal domiance matrices is given.Further, the relations among all generalized positive definite matrices are shown, also,some flaws and mistakes in the references are corrected.展开更多
A new approach that bounds the largest eigenvalue of 3 × 3 correlation matrices is presented. Optimal bounds by given determinant and trace of the squared correlation matrix are derived and shown to be more strin...A new approach that bounds the largest eigenvalue of 3 × 3 correlation matrices is presented. Optimal bounds by given determinant and trace of the squared correlation matrix are derived and shown to be more stringent than the optimal bounds by Wolkowicz and Styan in specific cases.展开更多
The symmetric,positive semidefinite,and positive definite real solutions of the matrix equation XA=YAD from an inverse problem of vibration theory are considered.When D=T the necessary and sufficient conditions fo...The symmetric,positive semidefinite,and positive definite real solutions of the matrix equation XA=YAD from an inverse problem of vibration theory are considered.When D=T the necessary and sufficient conditions for the existence of such solutions and their general forms are derived.展开更多
We shall give natural generalized solutions of Hadamard and tensor products equations for matrices by the concept of the Tikhonov regularization combined with the theory of reproducing kernels.
文摘We exploit the theory of reproducing kernels to deduce a matrix inequality for the inverse of the restriction of a positive definite Hermitian matrix.
基金The National Natural Science Foundation of China(No.11371089)the China Postdoctoral Science Foundation(No.2016M601688)
文摘The range and existence conditions of the Hermitian positive definite solutions of nonlinear matrix equations Xs+A*X-tA=Q are studied, where A is an n×n non-singular complex matrix and Q is an n×n Hermitian positive definite matrix and parameters s,t>0. Based on the matrix geometry theory, relevant matrix inequality and linear algebra technology, according to the different value ranges of the parameters s,t, the existence intervals of the Hermitian positive definite solution and the necessary conditions for equation solvability are presented, respectively. Comparing the existing correlation results, the proposed upper and lower bounds of the Hermitian positive definite solution are more accurate and applicable.
基金Supported by the Natural Science Foundation of Science and Technology Office of Guizhou Province (Grant No. J[2006]2002)
文摘By presenting a counterexample, the author of paper （ZHAO Li-feng. J. Math. Res. Exposition, 2007, 27（4）： 949-954） declared that some assertions in papers of LU Yun-xia, ZHANG Shu-qing （J. Math. Res. Exposition, 1999, 19（3）： 598-600）, HE Gan-tong （J. Math. Res. Exposition, 2002, 22（1）： 79-82） and YUAN Hui-ping （J. Math. Res. Exposition, 2001, 21（3）： 464-468） are wrong. In this note, we point out that the counterexample is wrong. Further discussion on these assertions and some related results are also given.
文摘This paper discusses the necessary and sufficient conditions for the existence of Hermite positive definite solutions of the quaternion matrix equation X<sup>m</sup>+ B*XB = C (m > 0) and its iterative solution method. According to the characteristics of the coefficient matrix, a corresponding algebraic equation system is ingeniously constructed, and by discussing the equation system’s solvability, the matrix equation’s existence interval is obtained. Based on the characteristics of the coefficient matrix, some necessary and sufficient conditions for the existence of Hermitian positive definite solutions of the matrix equation are derived. Then, the upper and lower bounds of the positive actual solutions are estimated by using matrix inequalities. Four iteration formats are constructed according to the given conditions and existence intervals, and their convergence is proven. The selection method for the initial matrix is also provided. Finally, using the complexification operator of quaternion matrices, an equivalent iteration on the complex field is established to solve the equation in the Matlab environment. Two numerical examples are used to test the effectiveness and feasibility of the given method. .
文摘For the lower bound about the determinant of Hadamard product of A and B, where A is a n × n real positive definite matrix and B is a n × n M-matrix, Jianzhou Liu [SLAM J. Matrix Anal. Appl., 18(2)(1997): 305-311]obtained the estimated inequality as follows det(A o B)≥a11b11 nⅡk=2(bkk detAk/detAk-1+detBk/detBk-1(k-1Ei=1 aikaki/aii))=Ln(A,B),where Ak is kth order sequential principal sub-matrix of A. We establish an improved lower bound of the form Yn(A,B)=a11baa nⅡk=2(bkk detAk/detAk-1+akk detBk/detBk-1-detAdetBk/detak-1detBk-1)≥Ln(A,B).For more weaker and practical lower bound, Liu given thatdet(A o B)≥(nⅡi=1 bii)detA+(nⅡi=1 aii)detB(nⅡk=2 k-1Ei=1 aikaki/aiiakk)=(L)n(A,B).We further improve it as Yn(A,B)=(nⅡi=1 bii)detA+(nⅡi=1 aii)detB-(detA)(detB)+max1≤k≤n wn(A,B,k)≥(nⅡi=1 bii)detA+(nⅡi=1 aii)detB-(detA)(detB)≥(L)n(A,B).
文摘In the paper,a necessary and sufficeent condition for generalized diagonal domiance matrices is given.Further, the relations among all generalized positive definite matrices are shown, also,some flaws and mistakes in the references are corrected.
文摘A new approach that bounds the largest eigenvalue of 3 × 3 correlation matrices is presented. Optimal bounds by given determinant and trace of the squared correlation matrix are derived and shown to be more stringent than the optimal bounds by Wolkowicz and Styan in specific cases.
文摘The symmetric,positive semidefinite,and positive definite real solutions of the matrix equation XA=YAD from an inverse problem of vibration theory are considered.When D=T the necessary and sufficient conditions for the existence of such solutions and their general forms are derived.
文摘We shall give natural generalized solutions of Hadamard and tensor products equations for matrices by the concept of the Tikhonov regularization combined with the theory of reproducing kernels.
