Use the factorization A = PDP-1 to compute Ak, where k represents an arbitrary positive integer....
part 1 and 2 please.
Let A = PDP-1 and P and D as shown below. Compute A4. (Simplify your answers.) Use the factorization A = PDP to compute AK, where k represents an arbitrary positive integer. -21-6912:1- Ak=
Prove that, for large integer k 〉 0, the 2-norm of an arbitrary matrix Ak behaves asymptotically like ー2+1 where j is the largest order of all diagonal submatrices J of the Jordan form with o(%)-ρ(A) and v is a positive constant. (Hint: refer to Greenbaum for an expression of the kth power of a j-by-j Jordan block)
For any two positive integers a, b, define k(a,b) to be the largest k such that a* | b but ak+1b. Given two positive integers x, y, show that (a) k(a, gcd(x, y)) = min{k(a, x), k(a, y)} for any positive integer a (b) k(a, lcm(z, y)) = max{k(a,a),k(a, y)} for any positive integer a. Hint: Think of the prime factorization of the numbers
For any two positive integers a, b, define k(a,b) to be the largest k such that...
sin ak 2. (1) Let k be a positive integer. Find the Laurent series expansion of f(x) = at z = 0 precisely (presenting a first few terms is not sufficient). (2) Find Res[f(x), 0). (3) Is the singularity of at z = O removable ? ਵ
Find the Fourier series off on the given interval. <x<0 OsX< F(x) = Give the number to which the Fourier series converges at a point of discontinuity of I. (if is continuous on the given interval, enter CONTINUOUS.) Let A = PDP-1 and P and D as shown below. Compute A Let A=PDP-1 and P and D A=1901 (Simplify your answers.) Use the factorization A = PDP-1 to compute Ak, where k represents an arbitrary integer. [x-» :)+(1:10:1 2:] Diagonalize...
4. Let A be a square matrix. Assume that Ak = 0 for some positive integer k. Then prove that a) 1-A is is invertible b) (1 - A)-1 = 1 + A + A + A + ................ + Ak-1 (This question is printed wrong in the text book, 10th edition. If you have this book, correct it)
A matrix A E Mnxn (F) is called nilpotent if, for some positive integer k, Ak O. A" O 1.Show that A eE Mnxn(F) is nilpotent the characteristic polynomial of A is t" 2. Show that if A, BE Mnxn(F) BA, then A + B is nilpotent. nilpotent and AB are 3. Show that if A, B e Mxn(F), A is nilpotent and AB BA, then AB is nilpotent. 4. If A E Mnxn(F) is nilpotent, find the inverse of...
Tems.] Use the second principle of induction to prove that every positive integer n has a factorization of the form 2m, where m is odd. (Hint: For n > 1, n is either odd or is divisible by 2.)
Problem 1 Let {ak} and {bk} be sequences of positive real numbers. Assume that lim “k = 0. k+oo bk 1. Prove that if ) bk converges, so does 'ak k=1 k=1 2. If ) bk diverges, is it necessary that ) ak diverges? k=1 k=1
One can compute an , where a > 0 and n is a positive integer in a brute-force way by repeatedly multiplying a: an = a