ECE-GY 5253 Final Fall 2022
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
ECE-GY 5253 Final
Fall 2022
Due: Dec. 19th. Monday, 7:30 pm, US Eastern Time.
1 Problem 1
Are the following statements true or false? If true, prove the statement. If false, give a counterexample.
1. Let A E R3×3. If each eigenvalue of A is zero, then A2 = 0. (Note: 0 denotes the zero matrix E R3×3)
2. Let A E Rn ×n be a real symmetric matrix, and λ1 ,λ2 ,...,λn the eigenvalues of A, then
IAI2 = mi(a)x|λi |
where |λi | denotes the absolute value of λi for i = 1,...,n.
3. Let A E Rn ×n and aij denote the element in ith row and jth column. If 0 ≤ aij < 1 Ai E n
{1,..., n}, j E {1,..., n}, and: aij < 1 for i = 1,...,n, then ρ(A) < 1 where ρ(A) is the spectral j=1
radius of A.
2 Problem 2
Let A = αβT where α,β E Rn and βT α 0.
1. When will the linear system ˙(x) = Ax be asymptotically or marginally stable?
2. Compute eAt .
2023-12-11