Several Matrix Inequalities

The main result of the first part of this thesis studies some special cases of Xingzhi Zhan's conjecture about unitarily invariant norms involving Hadamard product. The second part of this thesis partially solves Chi-Kwong Li's conjecture about the principal submatrices of the orthogonal projection matrices. The real symmetric matrices of a given order whose entries are in a given interval are very useful in control theory; the main result of the third part of this thesis gives an estimation of eigenvalues of the real symmetric matrices whose entries are in a given interval based on Professor Xingzhi Zhan's results. In the fourth part of this thesis we prove an upper bound for the spectral radius of the Hadamard product of nonnegative matrices and a lower bound for the minimum eigenvalue of the Fan product of M—matrices.