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A is mxn matrix Problem 7 (10pts) Prove any TWO of the following: Let A be...
Let A be an mx n matrix. Prove the following, using the appropriate transpose propertics and the definition of syneiric matrix: (a) AAT is a symmetric matrix. (b) ATA is a symmetric matrisx.
Problem 5. (10pts) Let A E Mm.n(k) be a matrix of rank 2. Prove that A can be written as B.C, where Be Mm.2(k) and CE M2.n(k).
How do you do this Linear Algebra problem? 6. Let A [ai i be an mxn matrix with RREF R-FF. Prove that i.. Tn there exists an m × m invertible matrix E such that аґ Eri for 1-i-n 6. Let A [ai i be an mxn matrix with RREF R-FF. Prove that i.. Tn there exists an m × m invertible matrix E such that аґ Eri for 1-i-n
Let A be an mx n matrix and B be an n xp matrix. (a) Prove that rank(AB) S rank(A). (b) Prove that rank(AB) < rank(B).
Let A be a matrix of size m xn. Show that AAT and AT A are both square matrices (equal number of rows and columns) (10 pts) If A is mXn then A is nXm so AA must have size mXm Similarly, A" A must be nxn
Prove that any m x n matrix A of rank k can be written as A = {k=1 u;v] where {u1, ..., Uk} and {V1, ... , Vk} are linearly independent sets. By SVD, any mxn matrix A of rank k can be written as A = {k=10;U;v] where {u1,..., Uk} and {V1, ... , Vk} are orthonormal sets and 01 > 02 > ... >0k > 0. For this problem, prove without using SVD.
Exercise 5. Let C be an mxn matrix. Assume that a is an n-vector that is independent of the rows of C. Let em+1 denote the last column of the identity matrix of order m +1, i.e., em+1 is an m + 1-vector of zeros, where the last entry is 1. If A is the (m + 1) x n matrix show that the system of equations Ag = em+1 is compatible. Note that we have made no assumptions about...
(a) Let A be a real n x m matrix. (i) State what conditions on n and m, if any, are needed such that the matrix AAT exists. Justify your statement. (ii) Assuming that the matrix AA exists, find its size. (iii) Assuming that the matrix AAT exists, prove using index notation that all diagonal elements of AAT are positive or equal to zero. (iv) Let 12 5 -3 A= 3-4 2 Calculate (AAT) -- (show all your working). 2)
Homework problem: Singular Value Decomposition Let A E R n 2 mn. Consider the singular value decomposition A = UEVT. Let u , un), v(1),...,v(m), and oi,... ,ar denote the columns of U, the columns of V and the non-zero entries (the singular values) of E, respectively. Show that 1. ai,.,a are the nonzero eigenvalues of AAT and ATA, v(1)... , v(m) the eigenvectors of ATA and u1)...,un) (possibly corresponding to the eigenvalue 0) are the eigenvectors of AAT are...
If Matrix A, r(A)=n, prove that r(AB)=r(B), for any B nxp, and show that for any invertible mxm matrix P, there exists Q mxn with full rank such that A=PQ