1 Problem 1 Are the following statements true or false? If true, prove the statement. If false, give a counterexample. 1. Let A € R³×3. If each eigenvalue of A is zero, then A² = 0. (Note: 0 denotes the zero matrix € R³x3) 2. Let A € Rnxn be a real symmetric matrix, and A₁, A2, ..., An the eigenvalues of A, then || A||2 = max |A₂| where |A₂| denotes the absolute value of λ; for i = 1, ..., n. 3. Let A € Rnxn and aij denote the element in ith row and th column. If 0 ≤ aij < 1 Vi € {1,...,n}, j = {1,..., n}, and Σ I j=1 aij < 1 for i = 1, ..., n, then p(A) < 1 where p(A) is the spectral radius of A.
1 Problem 1 Are the following statements true or false? If true, prove the statement. If false, give a counterexample. 1. Let A € R³×3. If each eigenvalue of A is zero, then A² = 0. (Note: 0 denotes the zero matrix € R³x3) 2. Let A € Rnxn be a real symmetric matrix, and A₁, A2, ..., An the eigenvalues of A, then || A||2 = max |A₂| where |A₂| denotes the absolute value of λ; for i = 1, ..., n. 3. Let A € Rnxn and aij denote the element in ith row and th column. If 0 ≤ aij < 1 Vi € {1,...,n}, j = {1,..., n}, and Σ I j=1 aij < 1 for i = 1, ..., n, then p(A) < 1 where p(A) is the spectral radius of A.
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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Please solve part 2 or 3 of q1 which will you can do 100% accurate
Transcribed Image Text:1 Problem 1
Are the following statements true or false? If true, prove the statement. If false, give a counterexample.
1. Let A € R³×3. If each eigenvalue of A is zero, then A² = 0. (Note: 0 denotes the zero matrix
€ R³x3)
2. Let A € Rnxn be a real symmetric matrix, and A₁, A2, ..., An the eigenvalues of A, then
||A||₂ = max |A₂|
1
where |A₁| denotes the absolute value of A₂ for i = 1,..., n.
3. Let A € Rxn and aij denote the element in th row and th column. If 0 ≤ aij < 1 Vi €
TL
{1,...,n}, j = {1,..., n}, and Σ aij < 1 for i = 1,..., n, then p(A) < 1 where p(A) is the spectral
radius of A.
2 Problem 2
Let A = aßT where a, B E R and B¹ a 0.
1. When will the linear system = Ar be asymptotically or marginally stable?
2. Compute et
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