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
<span style="line-height:1.5;"><span>In this paper, we consider a constrained low rank approximation problem: </span><img src="Edit_57d85c54-7822-4512-aafc-f0b0295a8f75.png" wi...<span style="line-height:1.5;"><span>In this paper, we consider a constrained low rank approximation problem: </span><img src="Edit_57d85c54-7822-4512-aafc-f0b0295a8f75.png" width="100" height="24" alt="" /></span><span style="line-height:1.5;"><span>, where </span><i><span>E</span></i><span> is a given complex matrix, </span><i><span>p</span></i><span> is a positive integer, and </span></span><span style="line-height:1.5;"></span><span style="line-height:1.5;"><span> is the set of the Hermitian nonnegative-definite least squares solution to the matrix equation </span><img src="Edit_ced08299-d2dc-4dbb-907a-4d8d36d2e87a.png" width="60" height="16" alt="" /></span><span style="line-height:1.5;"><span>. We discuss the range of </span><i><span>p</span></i><span> and derive the corresponding explicit solution expression of the constrained low rank approximation problem by matrix decompositions. And an algorithm for the problem is proposed and the numerical example is given to show its feasibility.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
We discuss two-stage iterative methods for the solution of linear systemAx = b, and give a new proof of the comparison theorems of two-stage iterative methodfor an Hermitian positive definite matrix. Meanwhile, we put...We discuss two-stage iterative methods for the solution of linear systemAx = b, and give a new proof of the comparison theorems of two-stage iterative methodfor an Hermitian positive definite matrix. Meanwhile, we put forward two new versionsof well known comparison theorem and apply them to some examples.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
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
From the formulas of the conjugate gradient, a similarity between a symmetric positive definite (SPD) matrix A and a tridiagonal matrix B is obtained. The elements of the matrix B are determined by the parameters of t...From the formulas of the conjugate gradient, a similarity between a symmetric positive definite (SPD) matrix A and a tridiagonal matrix B is obtained. The elements of the matrix B are determined by the parameters of the conjugate gradient. The computation of eigenvalues of A is then reduced to the case of the tridiagonal matrix B. The approximation of extreme eigenvalues of A can be obtained as a 'by-product' in the computation of the conjugate gradient if a computational cost of O(s) arithmetic operations is added, where s is the number of iterations This computational cost is negligible compared with the conjugate gradient. If the matrix A is not SPD, the approximation of the condition number of A can be obtained from the computation of the conjugate gradient on AT A. Numerical results show that this is a convenient and highly efficient method for computing extreme eigenvalues and the condition number of nonsingular matrices.展开更多
In this paper, nonlinear matrix equations of the form X ＋ A＊f1 （X）A ＋ B＊f2 （X）B = Q are discussed. Some necessary and sufficient conditions for the existence of solutions for this equation are derived. It is s...In this paper, nonlinear matrix equations of the form X ＋ A＊f1 （X）A ＋ B＊f2 （X）B = Q are discussed. Some necessary and sufficient conditions for the existence of solutions for this equation are derived. It is shown that under some conditions this equation has a unique solution, and an iterative method is proposed to obtain this unique solution. Finally, a numerical example is given to identify the efficiency of the results obtained.展开更多
For the expected value formulation of stochastic linear complementarity problem, we establish modulus-based matrix splitting iteration methods. The convergence of the new methods is discussed when the coefficient matr...For the expected value formulation of stochastic linear complementarity problem, we establish modulus-based matrix splitting iteration methods. The convergence of the new methods is discussed when the coefficient matrix is a positive definite matrix or a positive semi-definite matrix, respectively. The advantages of the new methods are that they can solve the large scale stochastic linear complementarity problem, and spend less computational time. Numerical results show that the new methods are efficient and suitable for solving the large scale problems.展开更多
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<span style="line-height:1.5;"><span>In this paper, we consider a constrained low rank approximation problem: </span><img src="Edit_57d85c54-7822-4512-aafc-f0b0295a8f75.png" width="100" height="24" alt="" /></span><span style="line-height:1.5;"><span>, where </span><i><span>E</span></i><span> is a given complex matrix, </span><i><span>p</span></i><span> is a positive integer, and </span></span><span style="line-height:1.5;"></span><span style="line-height:1.5;"><span> is the set of the Hermitian nonnegative-definite least squares solution to the matrix equation </span><img src="Edit_ced08299-d2dc-4dbb-907a-4d8d36d2e87a.png" width="60" height="16" alt="" /></span><span style="line-height:1.5;"><span>. We discuss the range of </span><i><span>p</span></i><span> and derive the corresponding explicit solution expression of the constrained low rank approximation problem by matrix decompositions. And an algorithm for the problem is proposed and the numerical example is given to show its feasibility.
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This work is supported by NSF of Shanxi province,20011041.
文摘We discuss two-stage iterative methods for the solution of linear systemAx = b, and give a new proof of the comparison theorems of two-stage iterative methodfor an Hermitian positive definite matrix. Meanwhile, we put forward two new versionsof well known comparison theorem and apply them to some examples.
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From the formulas of the conjugate gradient, a similarity between a symmetric positive definite (SPD) matrix A and a tridiagonal matrix B is obtained. The elements of the matrix B are determined by the parameters of the conjugate gradient. The computation of eigenvalues of A is then reduced to the case of the tridiagonal matrix B. The approximation of extreme eigenvalues of A can be obtained as a 'by-product' in the computation of the conjugate gradient if a computational cost of O(s) arithmetic operations is added, where s is the number of iterations This computational cost is negligible compared with the conjugate gradient. If the matrix A is not SPD, the approximation of the condition number of A can be obtained from the computation of the conjugate gradient on AT A. Numerical results show that this is a convenient and highly efficient method for computing extreme eigenvalues and the condition number of nonsingular matrices.
文摘In this paper, nonlinear matrix equations of the form X ＋ A＊f1 （X）A ＋ B＊f2 （X）B = Q are discussed. Some necessary and sufficient conditions for the existence of solutions for this equation are derived. It is shown that under some conditions this equation has a unique solution, and an iterative method is proposed to obtain this unique solution. Finally, a numerical example is given to identify the efficiency of the results obtained.
文摘For the expected value formulation of stochastic linear complementarity problem, we establish modulus-based matrix splitting iteration methods. The convergence of the new methods is discussed when the coefficient matrix is a positive definite matrix or a positive semi-definite matrix, respectively. The advantages of the new methods are that they can solve the large scale stochastic linear complementarity problem, and spend less computational time. Numerical results show that the new methods are efficient and suitable for solving the large scale problems.
