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Let 2 be an eigenvalue of A such that X₁ X2. Further let k be the unique integer such that and denote N((A − №₂1)-¹) ≤ N((A − X₁1)²) = N((A − λ₁1)*+¹), - W₂ = N((A-X₂1)*). λ21)^). (i) Prove that W₂ is (A - A₁I)-invariant. Hint: It might be useful to prove that (Z — nI)(Z — §I) = (Z − §I)(Z — nI), where Z is a square matrix Z and ŋ, are scalars. Then it follows that for any positive integer k we have (Z − nI)k (Z — §I) = (Z — §I)(Z — nI)k . (ii) Prove that if (X₁, x) is an eigenpair of A, then x Hint: Start by noting that x = 0 and (A - \₂1)x (X-μ)²-¹(A - µl)x. (iii) Combine (i) and (ii) to conclude that W₂ C U₁. W₂. 0. Also, note that (A - I)³x =

Let 2 be an eigenvalue of A such that X₁ X2. Further let k be the unique integer such that and denote N((A − №₂1)-¹) ≤ N((A − X₁1)²) = N((A − λ₁1)*+¹), - W₂ = N((A-X₂1)*). λ21)^). (i) Prove that W₂ is (A - A₁I)-invariant. Hint: It might be useful to prove that (Z — nI)(Z — §I) = (Z − §I)(Z — nI), where Z is a square matrix Z and ŋ, are scalars. Then it follows that for any positive integer k we have (Z − nI)k (Z — §I) = (Z — §I)(Z — nI)k . (ii) Prove that if (X₁, x) is an eigenpair of A, then x Hint: Start by noting that x = 0 and (A - \₂1)x (X-μ)²-¹(A - µl)x. (iii) Combine (i) and (ii) to conclude that W₂ C U₁. W₂. 0. Also, note that (A - I)³x =

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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Let 2 be an eigenvalue of A such that X₁ X2. Further let k be the unique integer such that and denote N((A − №₂1)-¹) ≤ N((A − λ₁1)²) = N((A − \₁1)*+¹), - N ((A — №21)^). W₂ = N (i) Prove that W₂ is (A - A₁I)-invariant. Hint: It might be useful to prove that (Z — nI)(Z — §I) = (Z − §I)(Z — nI), where Z is a square matrix Z and ŋ, are scalars. Then it follows that for any positive integer k we have (Z − nI)k (Z — §I) = (Z — §I)(Z — nI)k . (ii) Prove that if (A₁, x) is an eigenpair of A, then x W₂. Hint: Start by noting that x = 0 and (A - \₂1)x ‡ 0. Also, note that (A - I)³x = (X− μ)²-¹(A - µl)x. (iii) Combine (i) and (ii) to conclude that W₂ C U₁.
Transcribed Image Text:Let 2 be an eigenvalue of A such that X₁ X2. Further let k be the unique integer such that and denote N((A − №₂1)-¹) ≤ N((A − λ₁1)²) = N((A − \₁1)*+¹), - N ((A — №21)^). W₂ = N (i) Prove that W₂ is (A - A₁I)-invariant. Hint: It might be useful to prove that (Z — nI)(Z — §I) = (Z − §I)(Z — nI), where Z is a square matrix Z and ŋ, are scalars. Then it follows that for any positive integer k we have (Z − nI)k (Z — §I) = (Z — §I)(Z — nI)k . (ii) Prove that if (A₁, x) is an eigenpair of A, then x W₂. Hint: Start by noting that x = 0 and (A - \₂1)x ‡ 0. Also, note that (A - I)³x = (X− μ)²-¹(A - µl)x. (iii) Combine (i) and (ii) to conclude that W₂ C U₁.
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