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 >