Problem 8.4 An n × n matrix A is said to be idempotent if A2-A. (a)...
A matrix A is said to be idempotent if A2 - A. Show that each of the following is idempotent. 1 nJ H-X(XX)- I-H I-J
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....
(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...
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)
5. A matrix M is called idempotent if M2 = M. Which of the following statements must be true if M is an idempotent matrix? (i) M must be a square matrix. (ii) M must be either the zero matrix, or the identity matrix. (iii) M must be a invertible matrix. (iv) If M is an n xn matrix, then In - M must also be idempotent. (A) Only statements (i) and (ii) are true. (B) Only statements (i) and...
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) =
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.
solve 7th and 8th Find a 3-vector xo for which Axo Xo and Xo has length 1. b. Compute A"xo for n= 1,2.. 7. A square matrix A is said to be idempotent if AA- A. an canc cellation The a. Show that the following matrix is idempotent: 2.40 2-2 -4 (G3 -1 241) b. Show that if AB= A and BA- B, then A and B are both idempotent. Show that if A is idempotent, then A" = A...