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we have (A − 2)* = A* − λ,

we have (A − 2)* = A* − λ,

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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LEMMA 2.2. Let A be a normal transformation on a finite-dimensional inner product space X. Then if λ is an eigenvalue of A with eigenvector x, X is an eigen- value of A* with eigenvector x. Proof. Suppose x is an eigenvector of A and the corresponding eigenvalue which implies (A-2)x=0. By the results of Chapter 1 we have (A - λ)* = A* − λ, which, clearly, must commute with (A-2). Thus if A is normal, then so must be (A-2), which, by Lemma 2.1 implies that ||(A − 2)x|| = || (A − 2)*x|| = ||(A* —- )x|| = 0 ⇒(A* - X)x= 0 or A*x = 7x, or that x is an eigenvector of A* with corresponding eigenvalue X.
Transcribed Image Text:LEMMA 2.2. Let A be a normal transformation on a finite-dimensional inner product space X. Then if λ is an eigenvalue of A with eigenvector x, X is an eigen- value of A* with eigenvector x. Proof. Suppose x is an eigenvector of A and the corresponding eigenvalue which implies (A-2)x=0. By the results of Chapter 1 we have (A - λ)* = A* − λ, which, clearly, must commute with (A-2). Thus if A is normal, then so must be (A-2), which, by Lemma 2.1 implies that ||(A − 2)x|| = || (A − 2)*x|| = ||(A* —- )x|| = 0 ⇒(A* - X)x= 0 or A*x = 7x, or that x is an eigenvector of A* with corresponding eigenvalue X.
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