Problem. Let A=1-1-2-2-2 0-2 1 1 -1 2 1 0 (a) Find a Jordan form J for A. (b) Find the change of basis matrix X such that X AX -J
Problem. Let A=1-1-2-2-2 0-2 1 1 -1 2 1 0 (a) Find a Jordan form J for A. (b) Find the change of basis matrix X such that X AX -J
Problem. Let A=1-1-2-2-2 0-2 1 1 -1 21 0 (a) Find a Jordan form J for A (b) Find the change of basis matrix X such that X-1 AX = J.
Problem. Let A=1-1-2-2-2 0-2 1 1 -1 21 0 (a) Find a Jordan form J for A (b) Find the change of basis matrix X such that X-1 AX = J.
Let A = 4 0 0 2 1 2 1 2 1 (a)(4 marks) Find the eigenvalues of A. (b)(2 marks) Explain without any more calculations that A is diagonalisable. ((7 marks) Find three linearly independent eigenvectors of the matrix A. (d)(2 marks) Write an invertible matrix P such that -100 P-AP=0 40 0 0 3
1 1 14 -2 Problem 2.4. Let A 0 2 First find the eigenvalues of A. Then Pick one 0 0 -1 eigenvalue of A and find a basis for the eigenspace corresponding to the eigenvalue you chose.
ri 0 2-t] 3. Let Az = 0 t 1 .v= 10 0 2 (1) Find all possible t such that A, has determinant 1. (7 po (2) Find all possible t such that v is in the row space of Aų. (3) Find all possible t such that v is in an eigenspace of Al.
Exercise 24. Let 2 1 A =-1 3 1 0 -2 2 3 SDS-1. (i) Find a nonsingular matrix S and a diagonal matrix D such that A (ii) Find a matrix B that satisfies B2 = A
Exercise 24. Let 2 1 A =-1 3 1 0 -2 2 3 SDS-1. (i) Find a nonsingular matrix S and a diagonal matrix D such that A (ii) Find a matrix B that satisfies B2 = A
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.
linear algebra
1 1 2. Let A= 1 -1 2 0 -1 1 (a) Find the characteristic polynomial of A. You do not need to factor. (b) Verify that 71 0 4 is an eigenvector of A and identify the associated eigenvalue 11. 2 (C) Given that 12 = 2 is an eigenvalue of A, find a basis for its corresponding eigenbasis.
3. Let 13 0 A 2 1 -2 1 8 2 a) Find dim(rng(A). b) Find an equation relating the coordinates of a vector b - (br,b2,bs) in rng(A).
C3.)Let f(z,0) (1/θ)2(1-0)/0, 0 < x < 1,0 < θ < oo. . Find the maximurn likelihood estimator of θ. Show that the maximurn likelihood estimator is unbiased to θ