特殊矩阵和非线性矩阵方程的若干结果
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摘要
本文主要研究了以下内容：非线性矩阵方程Xs+A*X-tA=Q和Xs+A*F(X)A=Q的相关理论；增生-耗散矩阵范数不等式；两个非负矩阵Hadamard积的谱半径；两个M-矩阵Fan积的最小特征值；M-矩阵和M-矩阵逆的Hadamard积的最小特征值；对角魔方矩阵的性质分析；可对角化矩阵特征值的扰动界；任意矩阵的奇异值的扰动界.具体如下：
1.研究了非线性矩阵方程Xs+A*X-t=Q有Hermitian正定解的条件,其中Q是一个Hermitian正定矩阵.分以下四种情况讨论这个方程：
(a)当s和t都是正整数时,研究了上述非线性矩阵方程存在Hermitian正定解时[det(AA*)]1/n的上界；[detX]1/n的上下界；trX的上下界；Hermitian正定解特征值的上下界,特征值部分和的上下界以及特征值部分乘积的上界；
(b)当s≥1,0 (c)当0 (d)当s,t>0时,研究了Hermitian正定解的一些性质.非线性矩阵方程存在Hermitian正定解时A的谱半径和谱范数的估计.
2.对于非线性矩阵方程Xs+A*F(X)A=Q(s≥1),给出了此方程存在Hermitian(半)正定解与存在唯一解的条件,并由此得到唯一解的不动点迭代及其扰动分析.
3.给出了块增生-耗散矩阵中非对角块和对角块之间的范数不等式；给出了增生-耗散矩阵与其对角块之间的范数不等式.
4.给出了非负矩阵A与B Hadamard积的谱半径的上界；给出了非奇异M-矩阵A与B Fan积的最小特征值的下界,并给出了B与A-1Hadamard积的最小特征值的下界.
5.提出了对角魔方矩阵的定义,并证明了：对角魔方矩阵的秩小于等于2；对角魔方矩阵的子方阵为对角魔方矩阵；对角魔方矩阵与某类特定矩阵的乘积为对角魔方矩阵.
6.分别研究了可对角化矩阵特征值的扰动界与任意矩阵奇异值的扰动界.
In this dissertation, we study some problems on the nonlinear matrix equations Xs+A*X-tA=Q and Xs+A*F(X)A=Q; the norm inequalities for an accretive-dissipative matrix; the spectral radius of the Hadamard product of two nonnegative matrices; the minimum eigenvalue of the Fan product of two M-matrices; the minimum eigenvalue of the Hadamard product of M-matrices and the inverse of M-matrices; the properties of diagonally magic matrices; perturbation bounds for eigenvalues of diagonalizable matrices and perturbation bounds for singular values of arbitrary matrices. Our main results are as follows.
1. Investigations of the nonlinear matrix equation Xs+A*X-tA=Q, where Q is a Hermitian positive definite matrix. We consider four cases of this equation:
(a) s and t are positive integers, we present the upper bound of [det(AA*)]1/n when the nonlinear matrix equation Xs+A*X-tA=Q has a Hermitian positive def-inite solution. We also get the bounds of [detX]1/n and trX for the existence of a Hermitian positive definite solution. We obtain some bounds for the eigenval-ues of the Hermitian positive definite solution. We derive tight bounds of the partial sum and partial product about the eigenvalues of the solution X for the nonlinear matrix equation Xs+A*X-tA=Q;
(b) s≥1,0 (c)0 (d) s,t>0, we present some properties of the Hermitian positive definite solutions. We also get a property of the spectral radius of A for the existence of a solution. The spectral norm of A for the existence of a Hermitian positive definite solution is given.
2. Investigations of the nonlinear matrix equation Xs+A*F(X)A=Q with s≥1. Several sufficient and necessary conditions for the existence and uniqueness of the Hermitian positive (semidefinite) definite solution are derived; the fixed point itera-tion is given; and perturbation bounds are presented.
3. We present inequalities between the norm of off-diagonal blocks and the norm of diagonal blocks of an accretive-dissipative matrix, and inequalities between the norm of the whole accretive-dissipative matrix and the norm of its diagonal blocks.
4. A new upper bound on the spectral radius for the Hadamard product of A and B is obtained, where A and B are nonnegative matrices. Meanwhile, a new lower bound on the smallest eigenvalue for the Fan product of A and B is got, and some new lower bounds on the minimum eigenvalue for the Hadamard product of B and A-1are given, where A and B are nonsingular M-matrices.
5. A new class of matrices called diagonally magic matrices is presented and studied. In particular, we prove that every diagonally magic matrix has rank at most2; every square submatrix of a diagonally magic matrix is still a diagonally magic matrix; the products of diagonally magic matrices with some given matrices are diagonally magic matrices.
6. Perturbation bounds for the eigenvalues of diagonalizable matrices are derived. Per-turbation bounds for singular values of arbitrary matrices are also given.
1.研究了非线性矩阵方程Xs+A*X-t=Q有Hermitian正定解的条件,其中Q是一个Hermitian正定矩阵.分以下四种情况讨论这个方程：
(a)当s和t都是正整数时,研究了上述非线性矩阵方程存在Hermitian正定解时[det(AA*)]1/n的上界；[detX]1/n的上下界；trX的上下界；Hermitian正定解特征值的上下界,特征值部分和的上下界以及特征值部分乘积的上界；
(b)当s≥1,0
2.对于非线性矩阵方程Xs+A*F(X)A=Q(s≥1),给出了此方程存在Hermitian(半)正定解与存在唯一解的条件,并由此得到唯一解的不动点迭代及其扰动分析.
3.给出了块增生-耗散矩阵中非对角块和对角块之间的范数不等式；给出了增生-耗散矩阵与其对角块之间的范数不等式.
4.给出了非负矩阵A与B Hadamard积的谱半径的上界；给出了非奇异M-矩阵A与B Fan积的最小特征值的下界,并给出了B与A-1Hadamard积的最小特征值的下界.
5.提出了对角魔方矩阵的定义,并证明了：对角魔方矩阵的秩小于等于2；对角魔方矩阵的子方阵为对角魔方矩阵；对角魔方矩阵与某类特定矩阵的乘积为对角魔方矩阵.
6.分别研究了可对角化矩阵特征值的扰动界与任意矩阵奇异值的扰动界.
In this dissertation, we study some problems on the nonlinear matrix equations Xs+A*X-tA=Q and Xs+A*F(X)A=Q; the norm inequalities for an accretive-dissipative matrix; the spectral radius of the Hadamard product of two nonnegative matrices; the minimum eigenvalue of the Fan product of two M-matrices; the minimum eigenvalue of the Hadamard product of M-matrices and the inverse of M-matrices; the properties of diagonally magic matrices; perturbation bounds for eigenvalues of diagonalizable matrices and perturbation bounds for singular values of arbitrary matrices. Our main results are as follows.
1. Investigations of the nonlinear matrix equation Xs+A*X-tA=Q, where Q is a Hermitian positive definite matrix. We consider four cases of this equation:
(a) s and t are positive integers, we present the upper bound of [det(AA*)]1/n when the nonlinear matrix equation Xs+A*X-tA=Q has a Hermitian positive def-inite solution. We also get the bounds of [detX]1/n and trX for the existence of a Hermitian positive definite solution. We obtain some bounds for the eigenval-ues of the Hermitian positive definite solution. We derive tight bounds of the partial sum and partial product about the eigenvalues of the solution X for the nonlinear matrix equation Xs+A*X-tA=Q;
(b) s≥1,0
2. Investigations of the nonlinear matrix equation Xs+A*F(X)A=Q with s≥1. Several sufficient and necessary conditions for the existence and uniqueness of the Hermitian positive (semidefinite) definite solution are derived; the fixed point itera-tion is given; and perturbation bounds are presented.
3. We present inequalities between the norm of off-diagonal blocks and the norm of diagonal blocks of an accretive-dissipative matrix, and inequalities between the norm of the whole accretive-dissipative matrix and the norm of its diagonal blocks.
4. A new upper bound on the spectral radius for the Hadamard product of A and B is obtained, where A and B are nonnegative matrices. Meanwhile, a new lower bound on the smallest eigenvalue for the Fan product of A and B is got, and some new lower bounds on the minimum eigenvalue for the Hadamard product of B and A-1are given, where A and B are nonsingular M-matrices.
5. A new class of matrices called diagonally magic matrices is presented and studied. In particular, we prove that every diagonally magic matrix has rank at most2; every square submatrix of a diagonally magic matrix is still a diagonally magic matrix; the products of diagonally magic matrices with some given matrices are diagonally magic matrices.
6. Perturbation bounds for the eigenvalues of diagonalizable matrices are derived. Per-turbation bounds for singular values of arbitrary matrices are also given.
引文
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