Here I is a identity matrix
and H is a hat matrix.
Therefore I - H is an idempotent matrix.
Show each of the 3 following matrices is symmetric and idempotent ( J is a matrix with all 1 s) For the next few problems, let X = (X1X2), ßT = (β.β;), H the lat matrix for X, and Hi the hat matrix for X i. (I 1/nJ) ii. (I H)
Problem 8.4 An n × n matrix A is said to be idempotent if A2-A. (a) Show that the matrix is idempotent. (b) Show that if A is idempotent then the matrix (In-A) is also idempotent.
(1 point) A square matrix A is idempotent if A2 = A. Let V be the vector space of all 2 x 2 matrices with real entries. Let H be the set of all 2 x 2 idempotent matrices with real entries. Is H a subspace of the vector space V? 1. Does H contain the zero vector of V? choose 2. Is H closed under addition? If it is, enter CLOSED. If it is not, enter two matrices in...
1, and 6. An n xn matrix A is called idempotent if A2 = A. Some examples include lude [22] fool the identity In: Idempotents correspond to "projections onto a subspace," as we will discuss later. Prove the following statements: a) If A is idempotent then so is A". b) If A is idempotent, then so is In - A. c) If A and B are both idempotent, and AB = BA= Onxn (the zero matrix), then A+B is idempotent....
5) a) Suppose matrix A is idempotent. Show that A' must also be idempotent. b) Let A be an arbitrary 2x2 matrix. Show that the matrix AA' is symmetric (Again, to prove these results you cannot use specific examples.) 6) Let B I-A(A'A) A. a) Must B be square? Must A be square? Must (A'A) be square? b) Show that matrix B is idempotent. (Once again, do not use specific examples.)
5) a) Suppose matrix A is idempotent. Show that A' must also be idempotent. b) Let A be an arbitrary 2x2 matrix. Show that the matrix AA' is symmetric. Again, to prove these results you cannot use specific examples.)
Let A be an n x p matrix with n p. (a) Show that r(AA) = r(A). (b) Show that I - A(ATA) AT is idempotent. (c) Show that r(1-A(ATAYA") = n-r(A) Let A be an n x p matrix with n p. (a) Show that r(AA) = r(A). (b) Show that I - A(ATA) AT is idempotent. (c) Show that r(1-A(ATAYA") = n-r(A)
Please show work A and B are nxn matrices. If A is idempotent (that is, A2 = A), find all possible values of det(A). (Enter your answers as a comma-separated list.) det(A) =
Q2) Please show all working out neatly. If the answer is neat and correct I will upvote. Thanks! :) 2. Prove (without using Theorem 2.5) that if A and B are symmetric matrices, A + B is idempotent and AB = BA = 0, then both A and B are idempotent. (Hint: Use Theorem 2.4. Then derive two relations between the diagonalisations of A and B.) Theorem 2.4 Let A1, A2, ..., Am be a collection of symmetric k x...
7. Matrix A is said to be involutory if A2 = 1. Prove that a square matrix A is both orthogonal and involutory if and only if A is symmetric.