In [1, p. 35 exercise 2.14] we are asked to consider the solution to the set of linear equations
\mathbf{R}\mathbf{x}=\mathbf{-r} (1)

where \mathbf{R} is a n \times n hermitian matrix given by:
\mathbf{R}=P_1\mathbf{e_1}\mathbf{e_1}^H+P_2\mathbf{e_2}\mathbf{e_2}^H (2)

and \mathbf{r} is a complex n \times 1 vector given by
\mathbf{r}=P_1 e^{j2\pi f_1}\mathbf{e_1}+P_2 e^{j2\pi f_2}\mathbf{e_2} (3)

The complex vectors are defined in [1, p.22, (2.27)]. Furthermore we are asked to assumed that f_1=k/n, f_2=l/n for k,l are distinct integers in the range [-n/2,n/2-1] for n even and [-(n-1)/2,(n-1)/2] for n odd. \mathbf{x} is defined to be a n \times 1 vector. It is requested to show that \mathbf{R} is a singular matrix (assuming  n>2 ) and that there are infinite number of solutions. A further task is to find the general solution and also the minimum norm solution of the set of linear equations. The hint provided by the exercise is to note that \mathbf{e}_1/\sqrt{n},\mathbf{e}_2/\sqrt{n} are eigenvectors of \mathbf{R} with nonzero eigenvalues and then to assume a solution of the form
\mathbf{x}=\xi_1 \mathbf{e}_1 + \xi_2 \mathbf{e}_2 + \sum\limits_{i=3}^{n}\xi_i \mathbf{e}_i (4)

where \mathbf{e_i}^H\mathbf{e_1}=0, \mathbf{e_i}^H\mathbf{e_2}=0 for i=3,4,...,n and solve for \xi_1, \xi_2. read the conclusion >