| Estimations of the singular value, especially the lower bound for thesmallest singular value of matrix A, is an important subject of matrix analysis.While solving the linear equations by iterative method, we often need toestimate the spectral condition number of matrix A K(A)=σ1(A)/σn(A)≤(‖A‖1‖A‖∞)1/2/σn(A).Where the lower bound estimates of the smallest singular valueσn(A) isan important number. The lower bound ofσn(A) is also a very importantsubject for discussion in many other fields. So the lower bound estimates of thesmallest singular value have been getting much attention, and have importantvalue of theory and practicality.This thesis investigated some inequalities for singular value of matrix,lower bound estimates for the smallest singular value and the eigenvalues for sum of Hermitian matrix. This paper consists of four chapters.In chapter 1, the overview is given about the study at present in the world.In chapter 2, we give the singular value of nonsingular matrix A ordered as:σ1(A)≥σ2(A)≥L≥σn(A)＞0.Let 1≤k≤l≤n, by arithmetic-geometric mean inequality andσ12(A)+σ22(A)+L+σn2(A)=‖A‖F2,σ1(A)σ2(A)Lσn(A)=|det A|.We obtain some inequalities of the bounds of the singular valueσk(A)+L+σ1(A) andσk(A)Lσ1(A) as follows:Theorem2.3 Let A∈Cn×n(n≥3)is nonsingular, and 1≤k≤l≤n.Then 1/((l-k+1)1/2)(((k-1)/‖A‖F2)k-1|det A|2)1/2(n-k+1)≤σkl≤(‖A‖F2/l-(n/l-1)((l/‖A‖F2)l|det A|2)1/(n-l)1/2 (2.6)Theorem2.5 Let A∈Cn×n(n≥3)is nonsingular, and 1≤k≤l≤n-1.Then [(n-k+1)|det A|2(k/‖A‖F2)k](l-k+1)/2(n-k)≤σkLσ1 Theorem2.6 Let A∈Cn×n(n≥3)is nonsingular, and 1≤k≤l≤n. Then≤σkl≤1/|det A|(l+1/l)l+1/2(‖A‖F2/(n+1)n+1/2 (2.9)These inequalities involving k,l,n, det A and Frobenius norm only arepresented. Finally, we give some examples to show the effectiveness of the newinequalities.In chapter 3, base on det A and Frobenius norm of nonsingular matrix A,we first give the lower bound estimates of the smallest singular valueσn(A) ofnonsingular matrix A:σn＞(n-1/‖A‖F2)n-1/2|det A|]1+1/‖A‖F2(n-1/‖A‖F2)n-1|det A|2]n-1/2When‖A‖F2=n, we obtain conclusion as follows:σn＞(n-1/n)n-1/2|det A|[1+1/n(n-1/n)n-1|det A|2]n-1/2On the basis of the conclusion, let B=DA, where matrix A is nonsingular matrix and D=diag(1/r1(A),1/r2(A),L,1/rn(A)),we obtain another lower bound ofthe smallest singular value:σn(A)≥(n-1/n)n-1/2|det A|max{Cn(A)Sc, rn(A)Sr},where Sc=multiply from i=1 to n 1/ci(A)[1+1/n〔n-1/n〕n-1|det A|2〔multiply from i=1 to n〕1/ci(A)2]n-1/2,and Sr=multiply from i=1 to n 1/ri(A)[1+1/n〔n-1/n〕n-1|det A|2〔multiply from i=1 to n〕1/ri(A)2]n-1/2.Finally, we give some examples to show the effectiveness of the newbound.In chapter 4, we give two Hermite matrices A, B and their eigenvalues,and provide some inequalities of trAB. On the basis of the result, we obtain aninequality for sum of A,B.
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