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 tha...
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 n × n with AT-A. (The matrix A is syrnmetric.) Let B be 1 × n and let c E R. Define f : Rn → R by f(x) = 2.7, A . x + B . x + c. Show that The function f is a quadratic function Let A be n × n with AT-A. (The matrix A is syrnmetric.) Let B be 1 × n and let c E R. Define f : Rn...
(I) A square matrix E E M,xn(R) is idempotent if E-E. It is symmetric if E-E RR -[projyl& of projy relative to the standard basis (a) Let V C R be a subspace of R", and consider thé orthogonal projection projy onto V. Show that the representing matrix E & of IRn is both idempotent and symmetric. (b) Let E E Mnxn(R) be a matrix that is both idempotent and symmetric. Show that there is a subspace VCR" such that...
A square matrix E∈Mn×n(R) is idempotent if E2=E. It is symmetric if E = tE. (a) Let V⊆Rn be a subspace of Rn, and consider the orthogonal projection projV:Rn→Rn onto V. Show that the representing matrix E = [projV]EE of proj V relative to the standard basis E of Rn is both idempotent and symmetric. (b) Let E∈Mn×n(R) be a matrix that is both idempotent and symmetric. Show that there is a subspace V⊆Rn such that E= [projV]EE. [Hint: What...
(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...
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.
a) Let I be the n x n identity matrix and let O be the n × n zero matrix . Suppose A is an n × n matrix such that A3 = 0. Show that I + A is invertible and that (I + A)-1 = I – A+ A2. b) Let B and C be n x n matrices. Assume that the product BC is invertible. Show that B and C are both invertible.
A8.2 Let A be an m × n matrix and B be an n × p matrix. (a) Show that col(B) C null(A) if and only if AB = 0. (b) Show that if AB = 0, then rank(A) + rank(B) 〈 n. A8.2 Let A be an m × n matrix and B be an n × p matrix. (a) Show that col(B) C null(A) if and only if AB = 0. (b) Show that if AB = 0,...
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