6) Let A be a 5x5 matrix, with 3 different eigenvalues, and let i be an...
6) Let A be a 5x5 matrix, with 3 different eigenvalues, and let a be an eigenvalue of multiplicity 3. If A-21 has rank 2, is A defective? (Explain, as a yes or no answer will receive no credit).
6) Let A be a 5x5 matrix, with 3 different eigenvalues, and let 2 be an eigenvalue of multiplicity 3. If A-2/ has rank 2, is A defective? (Explain, as a yes or no answer will receive no credit).
(1 point) Suppose a 3 x 3 matrix A has only two distinct eigenvalues. Suppose that tr(A) = 1 and det(A) = 63. Find the eigenvalues of A with their algebraic multiplicities. The smaller eigenvalue has multiplicity and the larger eigenvalue has multiplicity
(b) The matrix B= 1 2 2 3 3 1 3 2 4 has eigenvalues 7,2, -1. i. Find a column and a row eigenvector of B corresponding to the Perron eigenvalue. ii. Find a rank one nonnegative matrix C such that the matrix B+C will have eigenvalues 13, 2, -1. iii. Let a and B be real numbers. Calculate the eigenvalues of D(a, b) = aB+ BC. iv. Find limno(+B)"
estion 3 Let A be an n x n symmetric matrix. Then, which of the following is not true? a) A is diagonalizable. b) If I is an eigenvalue of A with multiplicity k, then the eigenspace of has dimension k c) Some eigenvalues of A can be complex. d) All eigenvalues of A are real.
9. Let Q1 S Q2 be the eigenvalues of the matrix A and Qs 1 if Qi is a defective eigenvalue and Q. - 0 otherwise, where 86 100 01-81-94 T < 1 Let Q = 1013 + IQ1 + 2(Qal + 3(Qal). Then T = 5 sin2(100Q) satisfies:-(A) 0 (B) 1 T < 2.
9. Let Q1 S Q2 be the eigenvalues of the matrix A and Qs 1 if Qi is a defective eigenvalue and Q. - 0...
3 1. Let A = 0 (a) Compute the eigenvalues of A and specify their algebraic multiplicities. (b) For every eigenvalue 1, determine the eigenspace Ex and specify its dimension. (c) Is A a defective matrix? Why or why not? (d) Is A a singular matrix? Why or why not? (e) Determine the eigenvalues of (74) + 5.
2. Let A be an n x n real symmetric matrix or a complex normal matrix. Prove that tr(A) = X1 + ... + and tr(AⓇA) = 1212 + ... +14.12 where ....... An are the eigenvalues of A repeated with multiplicity (for example, if n = 3 and the eigenvalues of A are -3 and 7 but -3 has multiplicity 2 then 11 = -3, 12 = -3, and Az = 7). 3. Let A be an n x...
Let W be the 5x5 matrix from the data above, where each entry is a probability between 0 and 1 rather than a percentage: 0.76 0.03 0.18 0.02 0.01 0.04 0.85 0.11 0.00 0.00 W 10.10 0.03 0.80 0.04 0.03 0.07 0.01 0.15 0.700.07 0.10 0.03 0.00 0.050.82 PROBLeM 2.1. Observe that Wn- n where n-1. Explain why this makes sense. 0.26 0.16 PROBLEM 2.2. Observe that Wrp ~ p where p-0.38|. Explain why this makes sense. 0.09 0.11 PROBLeM...
6 8. Let A be a square matrix one of whose eigenvalues are 1. Is 12 an eigenvalue of B2. Why or why not?