Research On Matrix Trace Inequalities

In this thesis, we mainly study some inequalities on the trace of matrices,especially we prove some inequalities on the trace of positive semidefinite matrices.This thesis consists of four chapters, In the first chapter, some inequalities on the trace of nonnegative matrices and bounded linear operators on Hilbert space are discussed. The second chapter is the focus of this thesis. In this chapter, we prove some inequalities on the trace of positive semidefinite matrices.Let A be a positive semidefinite matix, B be a positive definite matix, and B-p/p+1∈C(A). Then, for any positive integer p, we have This result seems concise.In the third chapter, we prove a conjectured identity on the trace of special matrices with order 3.Let A bea 3×3 matrix with the real eigenvalues 1,λ,1/λ. Then, (1) (trA)2=trA2+2trA,(2)(trA)3+2trA3=3(trA)(trA2)+6.In the last chapter, some simple applications of the trace of matrices in control theory are studied.