sample covariance matrix

Question

Suppose two covariance function estimators, with the same formula except for a coefficient. Then make two sample covariance matrix(SCM) from each of the functions. Why should these matrices differ in negativeness or positiveness feature? To be clear $R=[R_{i-j}]$ should be non-negative definite(positive semi-definite) but $\tilde{R} =[\tilde R_{i-j}]$ is not necessarily non-negative definite. $$y(t)=\phi(t)\theta$$ $$R_{k}=\frac{1}{N}\sum_{t=1}^{N-k}y(t)y^T(t+k)\quad R_{-k}=R_k^T\quad k\ge 0$$ $$\tilde{R}_{k}=\frac{1}{N-k}\sum_{t=1}^{N-k}y(t)y^T(t+k)\quad \tilde R_{-k}=\tilde R_k^T\quad k\ge 0$$ The SCM is built by the following formulla. $$R=\begin{bmatrix}R_0&R_1&\cdots&R_{N-k}\\ R_1^T&R_0&\cdots& R_{N-k-1} \\ \vdots\\ R_{N-k}^T&\cdots&&R_0 \end{bmatrix}$$ They are used in system identification. main problem

Answer 1

For $N=2$ and $y(0)=(1,-1)/\sqrt{2}$ and $y(1)=(-1,-1)/\sqrt{2}$ you see that the first matrix has determinant 0 while the second one has determinant $>0$ so they have different signature.