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ECEN601_hw11_pfister

# ECEN601_hw11_pfister - ECEN 601 Assignment 11 Problems 1...

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ECEN 601: Assignment 11 Problems: 1. Show that if A is nonsingular, then AA H is positive deFnite. 2. Let F = 1 1 . . . 1 1 e - j 2 π/N . . . e - j 2 π ( N - 1) /N 1 e - j 4 π/N . . . e - j 4 π ( N - 1) /N . . . . . . . . . . . . 1 e - j 2 π ( N - 1) /N . . . e - j 2 π ( N - 1) 2 /N . The ( i, j )th element of this is f ij = e - 2 πij/N . ±or a vector x = [ x 0 , . . . , x n - 1 ] T , the product X = F x is the D±T of x . (a) Prove that the matrix F/ N is unitary. Hint: Show the following, N - 1 s n =0 e j 2 πnk/N = b N k = 0 mod N 0 k n = 0 mod N . (1) (b) A matrix C = c 0 c 1 c 2 . . . c N - 1 c N - 1 c 0 c 1 . . . c N - 2 . . . . . . . . . . . . . . . c 1 c 2 c 3 . . . c 0 is said to be a circulant matrix. Show that C is diagonalized by F , CF = F Λ, where Λ is diagonal. 3. Show that the minimum squared error when computing the LS b A x b b 2 2 solution is E 2 min = b U H 2 b b 2 2 . Interpret this result in light of the four fundamental subspaces. 4. Show that if a Hermitian matrix A is positive deFnite, then so is

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ECEN601_hw11_pfister - ECEN 601 Assignment 11 Problems 1...

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