Partial solution to last issue's homework assignment the direction-of-arrival problem: coming at you

71 We model the sensor measurements as x 1 (t) = As(t) + ∈ 1 (t), x 2 (t) = AΦs(t) + ∈ 2 (t). Consider the following recipe: • Find a matrix B of size d × m so that BA is d × d and full rank. • Find a matrix C of size n × d so that SC is d × d and full rank. • Find d vectors z k and d values λ k so that BAΦSCz k = λ k BASCz k. Problem 1. Show that the eigenvalues λ k are equal to the diagonal entries of Φ. Answer: Let w k = SCz k , and multiply the equation BAΦSCz k = λ k BASCz k by (BA) –1 to obtain Φw k = λ k w k , k = 1, …, d. By the definition of eigenvalue, we see that λ k is an eigen-value of Φ corresponding to the eigenvector w k. Because Φ is a diagonal matrix, its eigenvalues are its diagonal entries, so the result follows. Problem 2. Suppose that the singular value decomposition (SVD) of X is UΣW H , where σ i = 0, i > d. Let Σ 1 be the square diagonal matrix with entries σ 1 , …, σ d , and partition U into U = , where U 1 and U 2 have m rows and d columns, so that X 1 = AS = U 1 [Σ 1 , O d×(n–d) ]W H , X 2 = AΦS = U 2 [Σ 1 , O d×(n–d) ]W H , where O d×(n–d) is the zero matrix of size d × (n – d). Let ^ U = [U 1 ,U 2 ] have SVD T∆V H , and denote the leading d × d submatrix of ∆ by ∆ 1. Partition V = , so that V 1 and V 2 have dimension d × d. Let B =[∆ 1 –1 , O d×(m–d) ]T H and C = W. Show that the eigenvalues λ k that satisfy the equation V 2 H z k = λ k V 1 H z k are φ k .