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linear Let A, and P be 4x4 matrices. If 6 is an eigenvalue of A, 3...
Problem 4 a) Let A and B be nxn matrices with an eigenvalue for A and i an eigenvalue for B. Is + i necessarily an eigenvalue for A +B? Is di necessarily an eigenvalue for AB? If so, explain why. If not, come up with a counterex- ample. What if and i have the same eigenvector x? b) If A and B are row equivalent matrices, do they have the same eigenvalues? If so, explain why. If not, give...
1) Let A and B be nxn matrices. Show that if I is a nonzero eigenvalue of AB, then it is also an eigenvalue of BA.
(10](3) State the definition of eigenvalue. It begins: Dn: (eigenvalue) Let 7V V be a linear operator and 1 € R. A is an eigenvalue of TW [10(4) 13 5 5 GIVEN: A E M(3,1), A = -2 -1 -2 1 2 -1 0 the linear operator, T:M(3,1) - M(3,1), Tz = At and v = -1 EM(3,1) and v is an eigenvector of T. FIND: The eigenvalue, 1, of T associated with u.
6. For each of the following matrices A solve the eigenvalue problem. If A is diagonalizable, find a matrix P that diagonalizes A by a similarity transformation D-PlAP and the respective diagonal matrix D. If A is not diagonalizable, briefly explain why -1 4 2 (d) A-|-| 3 1 -1 2 2 -1 0 1 6 3 (a) A- (b)As|0 1 0| (c) A-1-3 0 11 -4 0 3
3. Let A and B be any nxn matrices. Suppose ū is an eigenvector of A and A+B with corresponding eigenvalues 1 and p. Show that ū is also an eigenvector for B and find an expression for its corresponding eigenvalue. [2]
Linear Algebra:Question 5 [10 points] If A, B, and C are 4×4 matrices; and det(A) = 4, det(B) = −5, and det(C) = −4 then compute: Question 5 [10 points] If A, B, and C are 4x4 matrices; and det(A) = 4, det(B) = -5, and det(C)=-4 then compute: det(2CT A-18-10-1BICI) = 0
Write a MIPS program to that will take two 4x4 matrices, and calculate their sum and product, using row major, and column major math (so this is actually 4 problems, but obviously they’re all pretty related). I generated two sample arrays to test 2 1 9 2 7 9 10 10 3 4 4 4 2 5 4 4 8 7 1 2 2 7 8 6 7 5 6 8 9 4 8 9 The output of your program...
1. Let F :V + V be a linear map, and let be a eigenvalue of F. Show that the set of all eigenvectors associated with is a subspace.
6. Let T P2 P be a linear transformation such that T P2P2 is still a linear trans formation such that T(1) 2r22 T(2-)=2 T(1) = 2r22 T(12 - )=2 T(x2x= 2r T(r2)2x (a) (6 points) Find the matrix for T in some basis B. Specify the basis that you use. (d) (4 points) Find a basis for the eigenspace E2. (b) (2 points) Find det(T) and tr(T') (e) (4 points) Find a basis = (f,9,h) for P2 such that...
A projection is a nonzero linear operator P such that P2-P. Let v be an eigenvector with eigenvalue λ for a projection P, what are all possible values of X? Show that every projection P has at least one nonzero eigenvector. A projection is a nonzero linear operator P such that P2-P. Let v be an eigenvector with eigenvalue λ for a projection P, what are all possible values of X? Show that every projection P has at least one...