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) Let j be a positive integer such that N (B¹) = N(Bi+¹). Prove that R(B¹³) ʼn N(B¹) = {0} . n g) Prove that there exists the smallest positive integer k such that R" = R(B) + N(BK).

) Let j be a positive integer such that N (B¹) = N(Bi+¹). Prove that R(B¹³) ʼn N(B¹) = {0} . n g) Prove that there exists the smallest positive integer k such that R" = R(B) + N(BK).

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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6. Let B € Rnxn be an arbitrary matrix. Our goal is to prove that there exists a positive integer k such that R" = R(Bk) + N(Bk). (3) (a) Prove that if B is a nonsingular matrix, then (3) holds with k = 1. Proved. Next we want to show that (3) holds for some k even if B is a singular matrix. (b) Prove that N (B²) ≤ N (B²+¹) and R(B²) 2 R(B²+¹), for l = 0, 1, 2,.... Here Bº := I. Proved. (c) Prove that there exists an integer j such that N(B³) = N(B¹+¹). Proved. Proved. (d) If N(B¹) = N(Bi+¹) for some positive integer j, then for the same integer j we have that R(B¹) = R(B³+¹). (e) If N(B¹) = N(Bi+¹) for some positive integer j, then the equality also holds for each t > j, that is, N(B¹) = N(B+¹) for each t>j. Proved. (f) Let j be a positive integer such that N(B¹) = N(B₁+¹). Prove that R(B¹) n N (B¹) = {0} . (g) Prove that there exists the smallest positive integer k such that R" = R(B¹) + N(B¹).
Transcribed Image Text:6. Let B € Rnxn be an arbitrary matrix. Our goal is to prove that there exists a positive integer k such that R" = R(Bk) + N(Bk). (3) (a) Prove that if B is a nonsingular matrix, then (3) holds with k = 1. Proved. Next we want to show that (3) holds for some k even if B is a singular matrix. (b) Prove that N (B²) ≤ N (B²+¹) and R(B²) 2 R(B²+¹), for l = 0, 1, 2,.... Here Bº := I. Proved. (c) Prove that there exists an integer j such that N(B³) = N(B¹+¹). Proved. Proved. (d) If N(B¹) = N(Bi+¹) for some positive integer j, then for the same integer j we have that R(B¹) = R(B³+¹). (e) If N(B¹) = N(Bi+¹) for some positive integer j, then the equality also holds for each t > j, that is, N(B¹) = N(B+¹) for each t>j. Proved. (f) Let j be a positive integer such that N(B¹) = N(B₁+¹). Prove that R(B¹) n N (B¹) = {0} . (g) Prove that there exists the smallest positive integer k such that R" = R(B¹) + N(B¹).
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