1. Suppose that A is a symmetric matrix with A A+ orome integer 1. Show that...
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.)
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.)
Show each of the 3 following matrices is symmetric and idempotent ( J is a matrix with all 1 s) For the next few problems, let X = (X1X2), ßT = (β.β;), H the lat matrix for X, and Hi the hat matrix for X i. (I 1/nJ) ii. (I H)
6) In econometrics we frequently encounter matrices that are both symmetric and idempotent. Such a matrix A has the properties A",4 and A#AA. Use these properties to show that OS a“ 1, where a" is the ith diagonal element of A. [7 points] 6) In econometrics we frequently encounter matrices that are both symmetric and idempotent. Such a matrix A has the properties A",4 and A#AA. Use these properties to show that OS a“ 1, where a" is the ith...
Please solve both parts of this question! I've stared at it for a long time without knowing how to approach it. (1) A square matrix E є м,xn(R) is idempotent if E-E. It is symmetric if -t E. (a) Let V C Rn be a subspace of R, and consider the orthogonal projection projy R" Rn onto V. Show that the representing matrix E = projy18 of proj v relative to the standard basis of IRn is both idempotent and...
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...
Let A be a symmetric idempotent matrix, i.e., A² = A. (a) Prove that the only possible eigenvalues of A are 0 and 1. (b) Prove that trace(A) = rank(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...
Use R programming to solve Q2. A matrix operator H(G; k) on a pxp symmetric matrix G (iy)- with a positive integer parameter k (k < p) yields another p×p symmetric matrix H = (hij 1 with i=k,j = k; (a) Use one single loop to construct the function H(G; k) in R (b) Generate a random matrix X of dimension 7x5, each element of which is id from N(0,1). Use the function H(G; k constructed in (a) to compute...
Consider the singular value decomposition (svd) of a symmetric matrix, A- UAU Show that for any integer, n, An-UNU. Argue that for a psd matrix A, there must exist a square root matrix, A-such that 1/2 1/2 A 1/2