MATRIX THEORY CHAPTER 6 3.PERTURBATION THEOREM Theorem 3.Let B,E Mn and A=B+E.Assume that B is normal.Let ((A)....n(A)}be the eigenvalues of A in some given order and let ((B),(B)}be the eigenvalues of B in some order.Then for any i(A),there erists some j(B)such that A(A)-入y(B引≤IEIl2 Theorem 4 (Hoffman-Wielandt).Let B,EE Mn and A=B+E.Assume that A and B are both eso时A in some give Then ∑Ao(4-A(BP≤E3=∑leP MATRIX THEORY - CHAPTER 6 3 3. Perturbation Theorem Theorem 3. Let B, E ∈ Mn and A = B + E. Assume that B is normal. Let {λ1(A), ..., λn(A)} be the eigenvalues of A in some given order and let {λ1(B), ..., λn(B)} be the eigenvalues of B in some order. Then for any λi(A), there exists some λj (B) such that |λi(A) − λj (B)| ≤ 9E 92 . Theorem 4 (Hoffman-Wielandt). Let B, E ∈ Mn and A = B + E. Assume that A and B are both normal. Let {λ1(A), ..., λn(A)} be the eigenvalues of A in some given order and let {λ1(B), ..., λn(B)} be the eigenvalues of B in some order. Then there exists a permutation σ(i) of the integers 1, 2, ..., n such that Xn i=1 |λσ(i)(A) − λi(B)| 2 ≤ kEk 2 2 = X i,j |eij | 2