(2) Find a matrix A such that P = A (ATA) AT is the projection matrix onto the null space of ſi 3 0 LO 0 1 -21 5 ]
For the 3×2 matrix A: a) Determine the eigenvalues of ATA, and confirm that your eigenvalues are consistent with the trace and determinant of ATA. b) Find an eigenvector for each eigenvalue of ATA. c) Find an invertible matrix P and a diagonal matrix D such that P-1(ATA)P = D. d) Find the singular value decomposition of the matrix A; that is, find matrices U, Σ, and V such that A = UΣVT. e) What is the best rank 1...
A is mxn matrix Problem 7 (10pts) Prove any TWO of the following: Let A be a mx n matrix. Then • (AA+)+ = AA+ and (A+A)+ = A+A • A+ = (ATA)+AT = AT (AAT)+ • A+ = (ATA)-IAT and A+A = In if rank(A) = n, • A+ = AT (AAT)-1 and AA+ = Im if rank(A) = m, • A+ = AT, if the columns of A are orthogonal, that is ATA=In
Find ATA and AAT for the following matrix. 4-1-4-6 31 2 -5 4 AAT = What do you observe? This answer has not been graded yet.
True or False? 1. If σ is a singular value of a matrix A, then σ is an eigenvalue of ATA Answer: 2. Every matrix has the same singular values as its transpose Answer: 3. A matrix has a pseudo-inverse if and only if it is not invertible. Answer: 4. If matrix A has rank k, then A has k singular values Answer:_ 5. Every matrix has a singular value decomposit ion Answer:_ 6. Every matrix has a unique singular...
Exercise 7. Show that every singular n × n matrix can be made non-singular by changing at most n of its entries. Give an example that actually requires n entry changes. Exercise 7. Show that every singular n × n matrix can be made non-singular by changing at most n of its entries. Give an example that actually requires n entry changes.
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)
Question 7 (1 point) Let A be an REF matrix of the augmented matrix for a system of linear equations. If the system has three variables and A has three non-zero rows, then the system has at least one solution. True False
Let A be a 2 x 2 matrix such that ATA equals: 21 For each item below determine if it can be calculated from the information provided. If so, answer the associated question and fill in the associated boxes. Otherwise, place an X in all associated boxes. 1. The minimum value of I Av I subject to the condition Ivi - 1. What is the square of this length? 3 2. The slope of the line which contains the shortest...
6. True or False: (a) An eigenvalue of the matrix A is a non-zero vector y such that Ac = Xū. (b) Let A be a 3 x 4 matrix. Then ker A is non-trivial. (e) Let A be an n x n matrix. Ta is injective (i.e. one-to-one) if and only if TA is surjective (i.e. onto). (d) If A is a singular matrix, then A must have an eigenvalue. (e) The set {A € M,(F): det(A) = +1}...