【作者】 张云；

【导师】 詹兴致；

【作者基本信息】 华东师范大学 ， 运筹学与控制论， 2014， 博士

【摘要】 本文研究了半正定分块矩阵、压缩矩阵、增生-耗散算子矩阵、非负矩阵的谱半径、矩阵的实部与虚部、矩阵的和与其绝对值的和、矩阵的项秩以及子空间之间的夹角与极小夹角的几个问题.本文的工作分为以下几部分：1.建立了半正定分块矩阵与各块子矩阵之间特征值(奇异值)的优超关系.这些优超关系推广了Furuichi与Lin, Turkmen, Paksoy和Zhang,以及Lin和Wolkowicz的结果.2.研究了严格压缩矩阵的奇异值不等式,给出了严格压缩矩阵的Lewent型奇异值的弱对数优超不等式.Lin的关于严格压缩矩阵的Lewent型行列式不等式是其中一种特殊情形.3.研究了增生-耗散算子矩阵的酉不变范数,回答了Lin和Zhou提出的一个公开问题.4.研究了非负矩阵的Hadamard积与普通乘积之间的谱半径的关系.首先给出Huang关于非负矩阵的谱半径不等式的新证明；其次证明了非负矩阵Hadamard积的Cauchy-Schwarz不等式；最后,将Audenaert关于非负矩阵的Hadamard积的谱半径的不等式推广到任意多个的情形,从而改进了Huang的不等式.5.指出了在Marshall, Olkin和Arnold的专著中关于矩阵实部的特征值的一个结果的漏洞,给出了它的修改后的版本.同时,给出关于矩阵实部和虚部的奇异值的不等式,并通过数值例子说明这些不等式中的常数因子是最佳的.6.研究了矩阵的和与其绝对值的和之间的酉不变范数不等式.我们给出了有限个正规矩阵的和与其绝对值的和的奇异值的弱对数优超关系.作为应用,推广了Zhan,Bourin和Uchiyama的相关结果.此外,我们证明了下面的结果：设A和B为复矩阵.则||A+B||≤2|||A|+|B|||1/2|||A|(?)|B|||1/2,对于任何酉不变范数‖·‖成立.该结果蕴涵了Lee的不等式.进一步地,我们通过数值例子说明该不等式中的常数因子(?)2是最佳的.7.研究了矩阵的项秩和非零元的个数之间的关系.确定了给定项秩的一般(0,1)-矩阵中含有1的个数,对称(0,1)-矩阵中含有1的个数和主对角线为零的对称(0,1)-矩阵中含有1的个数,并且确定了给定项秩中那些取到最多1的个数的矩阵.8.刻画了子空间之间的夹角等于极小夹角的那些子空间对.同时,刻画了那些子空间之间的夹角为π/2的子空间对.给出了J.K.Baksalary关于正交投子的可交换性结果的一个推广更多还原

【Abstract】 We study some problems on positive semidefinite block matrices, strict contractions, accretive-dissipative operators, nonnegative matrices, the real part and the imaginary part of matrices, the sums of matrices and the sums of the corresponding absolute values of matrices, term rank, the angles and the minimal angles between two closed subspaces of a complex Hilbert space. Our main results are as follows.1. We establish a majorization relation on the eigenvalues of a positive semidefinite block matrix and its blocks. These majorization inequalities generalize some results due to Furuichi and Lin, Turkmen, Paksoy and Zhang, Lin and Wolkowicz.2. We obtain a Lewent type weakly log-majorization relation on the singular values of strictly contractive matrices. A special case of this result is the Lewent type determinantal inequality due to Lin.3. We investigate unitarily invariant norm inequalities for accretive-dissipative operator matrices. An open problem raised by Lin and Zhou is solved.4. We first give a new proof of the spectral radius inequality for nonnegative matrices due to Huang. We then prove an inequality which may be regarded as a Cauchy-Schwarz inequality for spectral radius of nonnegative matrices. Finally, we refine Huang’s inequality on the spectral radius of nonnegative matrices, which generalizes Audenaert’s result to an arbitrary finite number of nonnegative matrices.5. We point out a gap on an eigenvalue inequality for real parts of matrices in a mono-graph by Marshall, Olkin and Arnold and we give a modified version of this inequal-ity. We obtain singular value inequalities for the real part and the imaginary part of matrices. Meanwhile, several examples show that the constant factors in these inequalities are best possible. 6. We study unitarily invariant norm inequalities between the sums of matrices and the sums of the corresponding absolute values. We obtain a weak log-majorization relation on the singular values of the sums of normal matrices and the sums of the corresponding absolute values. As applications, we generalize some results due to Zhan, Bouring and Uchiyama. In addition, we prove that if A, B are complex matrices, then||A+B||≤2||A|+||B|||1/2|(?)|B||1/2, holds for every unitarily invariant norm||·||, which implies a result due to Lee. We also show that the factor2is best possible.7, We determine the possible numbers of nonzero entries in a matrix with a given term rank in the generic case, the symmetric case and the symmetric case with O’s on the main diagonal respectively. The matrices that attain the largest number of nonzero entries are also determined.8. We characterize those pairs of subspaces of a complex Hilbert space whose angle and minimal angle are equal and give several characterizations of the pairs of sub-spaces with angle equal to π/2, which generalizes a result due to J.K. Baksalary on the commutativity of orthogonal projectors.更多还原