5) a) Suppose matrix A is idempotent. Show that A' must also be idempotent. b) Let...
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)
(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...
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...
3) Let u33 2, and wE-ls -1] 2 yes, compute them. If any of them is not defined, explain why not. b) Treating u, v', and w' as matrices, are the products v'u, and ne detined? If yes, compute them. If any of them is not defined, explain why not. 4) If A and B are mxn matrices and k and t are real numbers, prove that a) (A +B)k- Ak+ BAk b) A(k +t) Ak+ At Note that to...
(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...
2. Let A be any matrix and let B= AAT a. Use a 2x2 matrix A, to verify that B is symmetric. b. Write one-line proof to show that B is symmetric. Do not use part a. 3. Using Gaussian Elimination, solve the homogeneous system 2x1 + x2 – 3x3 = 0 - x2 - 4x2 + 3x3 = 0 2 1 -3 oli +3707 1-4 3lol 1-4 30
Problem 5 (a) Let A be an n × m matrix, and suppose that there exists a m × n matrix B such that BA = 1- (i) Let b є Rn be such that the system of equations Ax b has at least one solution. Prove that this solution must be unique. (ii) Must it be the case that the system of equations Ax = b has a solution for every b? Prove or provide a counterexample. (b) Let...
(a) Let S be a symmetric positive definite matrix and define a function | on R" by 1/2 xx Sx . Prove that this function defines a vector norm. Hint: Use the Cholesky decomposition. (b) Find an example of square matrices A an This shows that ρ(A) is not a norm. Note: there are very simple examples. d B such that ρ(A+B)>ρ(A) + ρ(8) (a) Let S be a symmetric positive definite matrix and define a function | on R"...