194 Chap. 4 Duality theory Exercise 4.26 Let A be a given matrix. Show that exactly one of the fo...
Exercise 5.10. Let P be the transition matrix of a Markov chain (Xt)120 on a finite state space Ω. Show that the following statements are equivalent: (i) P is irreducible and aperiodic (ii) There exists an integer r 0 such that for all i,je Ω, (88) (ii) There exists an integer r 20 such that every entry of Pr is positive.
(f) Let A be symmetric square matrix of order n. Show that there exists an orthogonal matrix P such that PT AP is a diagonal matrix Hint : UseLO and Problem EK〗 (g) Let A be a square matrix and Rn × Rn → Rn is defined by: UCTION E AND MES FOR THE la(x, y) = хтАУ (i) Show that I is symmetric, ie, 14(x,y) = 1a(y, x), if a d Only if. A is symmetric (ii) Show that...
Exercise 6 requires using Exercises 4 and 5. Exercise 4. Let a be any real number. Prove that the Euclidean translation Ta given by Ta(x, y)(a, y) is a hyperbolic rigid motion. *Exercise 5. Let a be a positive real number. Prove that the transformation fa: HH given by fa(x, y) (ax, ay) is a hyperbolic rigid motion Exercise 6. Prove that given any two points P and Q in H, there exists a hyperbolic rigid motion f with f(P)...
0.5 0 0 5. Let P 0.5 0.6 0.3represent the probability transition matrix of a Markov chain with three 0 0.4 0.7 states (a) Show that the characteristic polynomial of P is given by P-ÀI -X-1.8λ2 +0.95λ-0.15) (b) Verify that λι 1, λ2 = 0.5 and λ3 = 0.3 satisfy the characteristic equation P-λ1-0 (and hence they are the eigenvalues of P) c) Show thatu3u2and u3are three eigenvectors corresponding to the eigenvalues λι, λ2 and λ3, respectively 1/3 (d) Let...
a 0 0 where a b, and c are positive numbers. Let S be the unit ball whose bounding surface has the equation x-x R3 + R3 be a linear transformation determined by the matrix A= 1 Complete Let 0 b 0 + x 0 0 c parts a and b below. u1 x1 2 ,2 2 a Show that T S is bounded by the ellipsoid with the equation 1 Create a vector u = that is within set...
In this exercise you will work with LU factorization of an matrix A. Theory: Any matrix A can be reduced to an echelon form by using only row replacement and row interchanging operations. Row interchanging is almost always necessary for a computer realization because it reduces the round off errors in calculations - this strategy in computer calculation is called partial pivoting, which refers to selecting for a pivot the largest by absolute value entry in a column. The MATLAB...
4. Exercise Let X, Y be RVs. Denote E[X] = Hy and E[Y] =py. Suppose we want to test the null hypothesis Ho : Mx = uy against the alternative hypothesis Hi : 4x > uy. Suppose we have i.i.d. pairs (X1,Yı),...,(Xn, Yn) from the joint distribution of (X,Y). Further assume that we know the X - Y follows a normal distribution. (i) Show that exactly) T:= (X-Y)-(ux-uy) - tn-1), Sin (3) where s2 = n-1 [?-,((X; – Y;) –...
Please all thank you Exercise 25: Let f 0,R be defined by f(x)-1/n, m, with m,nENand n is the minimal n such that m/n a) Show that L(f, P)0 for all partitions P of [0, 1] b) Let mE N. Show that the cardinality of the set A bounded by m(m1)/2. e [0, 1]: f(x) > 1/m) is c) Given m E N construct a partition P such that U(f, Pm)2/m. d) Show that f is integrable and compute Jo...
Please show all work in READ-ABLE way. Thank you so much in advance. Problem 2.2 n and let X ε Rnxp be a full-rank matrix, and Assume p Note that H is a square n × n matrix. This problem is devoted to understanding the properties H Any matrix that satisfies conditions in (a) is an orthogonal projection matriz. In this problem, we will verify this directly for the H given in (1). Let V - Im(X). (b) Show that...
Exercise 4 (Paired test, known normality of the difference). Let X, Y be RVs. Denote E[X] = 4x and E[Y] = uy. Suppose we want to test the null hypothesis Houx = My against the alternative hy- pothesis H ux #uy. Suppose we have i.i.d. pairs (X1,Y),...,(X,Y) from the joint distribution of (X,Y). Further assume that we know the X - Y follows a normal distribution. (i) Noting that X1 - Y1,..., Xn-Ynare i.i.d. with normal distribution, show that (exactly)...