J. Fix a positive integer N> 1 and consider the NX N matrix: = [ ON-1], (1) ej (2/N)k (N-1)] where the Nx1 column vectors ok, k = 0,..., N-1 are such that k = [1 (2/N)k (a) (1 point) Write down the matrix for N = 4. (Write all the 16 entries). (b) (1 point) Find complex numbers ,..., EN such that the entries of the matrix (for arbitrary N) satisfy the following property: []m=xm m, l = 1,... N. 2.m 1 Note: Matrices that possess this property (that is, their rows form a geometric progression) are known as Vandermonde matrices and have several nice properties. (c) (3 points) Prove that is a diagonal matrix. (Note: denotes the conjugate transpose of the matrix. The conjugate transpose of a m x n complex-valued matrix A Cmxn is denoted by AH Cmxn and is defined such that its entries [A]ke k = 1,..., m, l = 1,...,n satisfy [A]k. = ([Alek)". In particular, if A Rmxn (i.e. it has only real entries), then A = AT, i.e. the conjugate transpose is simply the transpose matrix. Hint: You might need to recall the geometric series formula.) (d) (1 point) What is the inner (dot) product (ok, e) for k l {0, 1,..., N-1}. What do you conclude about the vectors do. 1., ON-1? (Note: Recall here the inner product is defined in the conventional way of dot product of vectors, that is (ok, de) End [K] ([e]n)*). -1 n=0 =