6. Show that if A is an n x n symmetric matrix and B is an n x m matrix. Show that BT AB, BT B and BBT are symmetric matrices (10 pts)
Show that, given an m x n matrix A, there exists a unique matrix B satisfying the following properties: 1. AB and BA are symmetric. 2. ABA = A 3. BAB = B
(6) Let A denote an m x n matrix. Prove that rank A < 1 if and only if A = BC. Where B is an m x 1 matrix and C is a 1 xn matrix. Solution (7) Do the following: (a) Use proof by induction to find a formula for for all positive integers n and for alld E R. Solution ... 2 for all positive (b) Find a closed formula for each entry of A" where A...
(a) Suppose A is an n x n real matrix. Show that A can be written as a sum of two invertible matrices. HINT: for any le R, we can write A = XI + (A - XI) (b) Suppose V is a proper subspace of Mn,n(R). That is to say, V is a subspace, and V + Mn.n(R) (there is some Me Mn,n(R) such that M&V). Show that there exists an invertible matrix M e Mn,n(R) such that M&V....
2. (a) (10 marks) Suppose A is an n x n real matrix. Show that A can be written as a sum of two invertible matrices. HINT: for any l ER, we can write A = \I + (A – XI) (b) (10 marks) Suppose V is a proper subspace of Mn,n(R). That is to say, V is a subspace, and V + Mn,n(R) (there is some Me Mn,n(R) such that M&V). Show that there exists an invertible matrix M...
2. (a) (10 marks) Suppose A is an n x n real matrix. Show that A can be written as a sum of two invertible matrices. HINT: for any l ER, we can write A = \I + (A – XI) (b) (10 marks) Suppose V is a proper subspace of Mn,n(R). That is to say, V is a subspace, and V + Mn,n(R) (there is some Me Mn,n(R) such that M&V). Show that there exists an invertible matrix M...
In the context of multiple regression, define the n X n matrix M =- X(X'X)-'X'. (i) Show that M is symmetric and idempotent. (ii) Prove that m, the diagonals of the matrix M, satisfy 0 sm s 1 for t = 1, 2, ..., n. (iii) Consider the linear model y = XB + u satisfies the Gauss-Markov Assumptions. Let û be the vector of OLS residuals. Show that Eſûù' x) = oʻM (iv) Conclude that while the errors {u:...
In the context of multiple regression, define the n X n matrix M =- X(X'X)-'X'. (i) Show that M is symmetric and idempotent. (ii) Prove that m, the diagonals of the matrix M, satisfy 0 sm s 1 for t = 1, 2, ..., n. (iii) Consider the linear model y = XB + u satisfies the Gauss-Markov Assumptions. Let û be the vector of OLS residuals. Show that Eſûù' x) = oʻM (iv) Conclude that while the errors {u:...
2. (a) (10 marks) Suppose A is an n x n real matrix. Show that A can be written as a sum of two invertible matrices. HINT: for any XER, we can write A= XI + (A - XI) (b) (10 marks) Suppose V is a proper subspace of Mn.n(R). That is to say, V is a subspace, and V #Mnn(R) (there is some Me M.,n(R) such that M&V). Show that there exists an invertible matrix M e Mn.n(R) such...
Show that the cost of QR is approximately 2mn2 flops where A is an m x n matrix. [Note: This is a numerical linear algebra problem].