3. Let A and B be any nxn matrices. Suppose ū is an eigenvector of A...
Let А and B be similar nxn matrices. That is, we can write A = CBC- for some invertible matrix с Then the matrices A and B have the same eigenvalues for the following reason(s). A. Both А and A. Both А and B have the same characteristic polynomial. B. Since A = CBC-1 , this implies A = CC-B = IB = B and the matrices are equal. C. Suppose that 2 is an eigenvalue for the matrix B...
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...
QUESTIONS Problem 3. Let P, Q be nxn matrices with PQ = QP. Suppose that is nonsinsingular and veR" is a nonzero eigenvector of P. Determine which of the following statements is True. e: v and Qü are eigenvectors of P with the same eigenvalues. 12: v and Qü are eigenvectors of P with distinct eigenvalues. T: Qü is not an eigenvector of P. V: None of the other answers. e оооо
a) suppose that the nxn matrix A has its n eigenvalues arranged
in decreasing order of absolute size, so that >>....
each eigenvalue has its corresponding eigenvector, x1,x2,...,xn.
suppose we make some initial guess y(0) for an eigenvector.
suppose, too, that y(0) can be written in terms of the actual
eigenvectors in the form y(0)=alpha1.x1 +alpha2.x2
+...+alpha(n).x(n), where alpha1, alpha2, alpha(n) are constants.
by considering the "power method" type iteration y(k+1)=Ay(k) argue
that (see attached image)
b) from an nxn...
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.
4. (a) (6 marks) Let A be a square matrix with eigenvector v, and corresponding eigenvalue 1. Let c be a scalar. Show that A-ch has eigenvector v, and corresponding eigenvalue X-c. (b) (8 marks) Let A = (33) i. Find the eigenvalues of A. ii. For one of the eigenvalues you have found, calculate the corresponding eigenvector. iii. Make use of part (a) to determine an eigenvalue and a corresponding eigenvector 2 2 of 5 - 1
SOLVE BOTH 4 and 5!!
4. Let A and B be two nxn matrices. Suppose that AB is invertible. Show that the system Ar 0 has only the trivial solution 5. Given that B and D are invertible matrices of orders n and p respectively, and A = Find A by writing A as a suitably partitioned matrix
5. Let A, B E Mmxm(R) and let v be an eigenvector of A with eigenvalue 1, and v be an eigenvector of B with eigenvalue j. (a) Show that v is an eigenvector of AB. What is the corresponding eigenvalue? (b) Show that v is an eigenvector of A+B. What is the corresponding eigenvalue?
True False a) For nxn A, A and AT can have different eigenvalues. b) The vector v 0 cannot be an eigenvector of A. c) If λ's an eigenvalue of A, then λ2 is an eigenvalue of A2. True False d) If A is invertible, then A is diagonalizable. e) If nxn A is singular, then Null(A) is an eigenspace of A. f) For nxn A, the product of the eigenvalues is the trace of A. True False g) If...
2 0 -21 3. Let A= 1 3 2 LO 0 3 (a) Find the characteristic equation of A. in Find the other (b) One of the eigenvalues for A is ) = 2 with corresponding eigenvector 1 10 eigenvalue and a basis for the eigenspace associated to it. (e) Find matrices S and B that diagonalize A, if possible.