x n matrices, then A is similar to B if and only if Problem 15 [10...
Problem 5. Let n N. The goal of this problem is to show that if two real n x n matrices are similar over C, then they are also similar over IK (a) Prove that for all X, y є Rnxn, the function f(t) det (X + ty) is a polynomial in t. (b) Prove that if X and Y are real n × n matrices such that X + ừ is an invertible complex matrix, then there exists a...
Two n x n matrices A and B are called similar if there is an invertible matrix P such that B = P-AP. Show that two similar matrices enjoy the following properties. (a) They have the same determinant. (b) They have the same eigenvalues: specifically, show that if v is an eigenvector of A with eigenvalue 1, then P-lv is an eigenvector of B with eigenvalue l. (c) For any polynomial p(x), P(A) = 0 is equivalent to p(B) =...
oru 2 Let A and B be two n x n matrices. There exists a nonsingular matrix P such that PB = AP. Then which of the following is always true? a) A and B are not similar b) A and B have the same eigenvalues c) A does not have any characteristic polynomial d) B does not have any characteristic polynomial
For the following six questions, indicate whether the following statements are true or false. In each case give a reason for your answer. Problem 13 [10 pts) If L:V +W is a linear transformation of vector spaces and U CW is a subspace of W, then {v € V | L(v € U} CW is a subspace of V. Problem 14 (10 pts) The set {A E R2x2 | A is nonsingular} is a subspace of R2x2. Problem 15 (10...
let a and b be n*n similar matrices, namely, B=S^-1 AS. show that the matrices a and b have the same characteristic polynomial, det(a-λI)=det(b-λI) and, consequently, the same eigenvalues.
Problem 16 (10 pts) For an n x n matrix A, pa(t) = t.q(t) for some polynomial q(t) precisely when Det(A) = 0. Problem 17 (10 pts) If W CR” is a subspace and v eR”, then pw(v) is the least-squares approximation to v by a vector in W except when pw(v) = 0. Problem 18 (10 pts) If A is a real n xn matrix, then the pairing defined by <v, w >:=yT * AT * A* w is...
Problem 16 (10 pts) For an n x n matrix A, PA(t) = t.q(t) for some polynomial q(t) precisely when Det(A) = 0. Problem 17 (10 pts) If W CR” is a subspace and ve R", then pw(v) is the least-squares approximation to v by a vector in W except when pw(v) = 0. Problem 18 (10 pts) If A is a real n x n matrix, then the pairing defined by <v, w >:=yT * AT * A *W...
Help! Let B and C be similar nxn matrices. Prove that the matrices given by: I +5B - 2B4 and I +5C - 204 are similar. (6 pts)
T F Matrices with the same eigen values must be similar T F Similar matrices must have the same eigen vectors T F Similar matrices must have the same eigen values T F Similar matrices must have the same characteristic polynomial T F If A and B are similar, then they must be invertible T F If A and B are similar and are both invertible, then A^(-1) is similar to B^(-1)
A and B are n* n matrices. show that (b) Show that if there is a 1 ER such that A - 1I is similar to B - 11, then A is similar to B.