Problem 3. Let X ∼ Nn(µ, V ). Find the distribution of wTX if w is an eigenvector of V with eigenvalue λ.
Problem 3. Let X ∼ Nn(µ, V ). Find the distribution of wTX if w is...
Let V be a vector space, let S, T L(V), and assume that ST = TS. Prove that if ˇ V is an eigenvector for T with eigenvalue λ, then λ is also an eigenvalue for S Find an eigenvector for λ with respect to S, and prove your answer is correct. Let V be a vector space, let S, T L(V), and assume that ST = TS. Prove that if ˇ V is an eigenvector for T with eigenvalue...
Let AA be an n×nn×n matrix. Prove that if x⃗ x→ is an eigenvector of AA corresponding to the eigenvalue λλ, then x⃗ x→ is also an eigenvector of A+cIA+cI, where cc is a scalar. Moreover, find the corresponding eigenvalue of A+cIA+cI.
7.3 (Eigenvalues II) Let V be a vector space over K and let f,g E End(V). Show that: a) If-1 is an eigenvalue of ff, then 1 is an eigenvalue of f3. b) If u is an eigenvector off o g to the eigenvalue λ such that g(v) 0, then g(v) is an eigenvector of g o f. If, in addition, dim V < oo,then f o g and go f have the same eigenvalues c) If {ul, unt is...
A projection is a nonzero linear operator P such that P2-P. Let v be an eigenvector with eigenvalue λ for a projection P, what are all possible values of X? Show that every projection P has at least one nonzero eigenvector. A projection is a nonzero linear operator P such that P2-P. Let v be an eigenvector with eigenvalue λ for a projection P, what are all possible values of X? Show that every projection P has at least one...
2. Let A be an invertible n x n matrix, and let (v) E C be an eigenvector of A with corresponding eigenvalue X E C. (a) Show that +0. (b) Further show that v) is also an eigenvector of A- with corresponding eigenvalue 1/1.
Problem 5, Show that if λ is an eigenvalue of a matrix A and v is the corresponding eigenvector, then eAtv is a solution of the ODE X AX.
Problem 2 Is λ 3 an eigenvalue of 13-2 / ? If so, find a corresponding eigenvector.
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
2. (10 points) Suppose v is an eigenvector of A with eigenvalue X, and let c be a real number. Show that v is an eigenvector of A+cI, where I is the appropriately sized identity matrix. What is the corresponding eigenvalue?
Problem V: Given v(x) = and w (x) and w (x) = 10x + 3, find (Vow)(x) and simplify. Question 5