9.1 (Characteristic polynomial) Let A E Knxn and ФА : Knxn → Knxn defined by ФА (X):-AX. Show tha...
Let k be a field of positive characteristic p, and let f(x)be an irreducible polynomial. Prove that there exist an integer d and a separable irreducible polynomial fsep (2) such that f(0) = fsep (2P). The number p is called the inseparable degree of f(c). If f(1) is the minimal polynomial of an algebraic element a, the inseparable degree of a is defined to be the inseparable degree of f(1). Prove that a is inseparable if and only if its...
5. Let A € Mnxn(C) with characteristic polynomial p(x) = cxºII-1(d; – x) and li + 0, Vi, a E Z>o. Show that if dim(ker(A))+k=n, then A= C2 for some complex matrix C.
Problem 2. (a) Let A be a 4 x 4 matrix with characteristic polynomial p(t) = +-12+} Find the trace and determinant of A. 2 e: tr(4) and det(A) = 0 12: tr(A) = 0 and det(A) 2 3 2 T: tr(A) = 0 and det(A) 3 : None of the other answers 01 OW
Algebra 2 -1 - Let A 1 2 -1 -1 -1 2 The characteristic polynomial of A is X(A - 3)2. (a) Find the eigenspaces of A and verify that the dimension of each eigenspace is equal to the multiplicity of the corresponding eigen value (b) Write down a matrix P that orthogonally diagonalises A You must show all your working Algebra 2 -1 - Let A 1 2 -1 -1 -1 2 The characteristic polynomial of A is X(A...
3. Let Z= (3 a 2 x 2 matrix over Zs. Find the characteristic polynomial of Z and determine for which values of h e Z5, Z is diagonalizable.
(i) Show that if X, 9X then aX+b aX + b for any real a and b. (ii) If X has zero mean and variance op, show that P(X 2 t) 2, fort > 0 (iii) Show that X, 0 if and only if EG .) +0, as n +. (iv) Let X1 X2, ... be independent and identically distributed random variables whose com- mon characteristic function satisfies '0) = in. Show that x;".
11. (adapted from 1.6 8) Prove the characteristic polynomial of matrix A - is p(x) = 12 - (a+d)X + ad-bc = 0. Show that p(A) = A - (a + d) A+ (ad - bc)1 = 0. 12. (adapted from 1.6 14) Suppose A has eigenvalues 0,0,3 with independent eigenvectors u, v,w. (a) Give the vectors span the nullspace and the column space. (b) Find a particular solution to Ax=w. Find all solutions. (c) Does w + u in...
3. Let f: RP-R (a) If f(x)-Ax + b, x E R A є Mq.p and b є R9, show that f is p. where differentiable everywhere and calculate its total derivative (b) If f is differentiable everywhere and Df (x)A, for some A E Mp and all q.p x E Rp, show that there exists b E R, such that f(x) = Ax + b for all x E Rp 3. Let f: RP-R (a) If f(x)-Ax + b,...
Algebra 2 -1 - Let A 1 2 -1 -1 -1 2 The characteristic polynomial of A is X(A - 3)2. (a) Find the eigenspaces of A and verify that the dimension of each eigenspace is equal to the multiplicity of the corresponding eigen value (b) Write down a matrix P that orthogonally diagonalises A You must show all your working
Problem 4 Let V be the vector space of functions of the form f(x) = e-xp(x), where p(x) is a polynomial of degree (a) Find the matrix of the derivative operator D = d/dx : V → V in the basis ek = e-xXk/k!, k = 0, 1, . .. , n, of V. (b) Find the characteristic polynomial of D. (c) Find the minimal polynomial of D n. Problem 4 Let V be the vector space of functions of...