Matrix Inequalities,Term Rank And Angles Between Subspaces

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.