Recall that covariance matrix is a symmetric positive-definite square matrix of the form,
\[\Sigma = \begin{pmatrix} \sigma_{11} & \sigma_{12} \\ \sigma_{21} & \sigma_{22} \end{pmatrix}\]where each element is computed via the following equation,
\[\sigma_{ij} = \sum_1^{n} ~ (x_i-\overline{x})(y_j-\overline{y})\]where $n$ represents the number of data points and $x_i$ and $x_j$ are individual data points. Also, recall that the corresponding correlation matrix to this covariance matrix of data is defined as the following,
\[\text{Cor} = \begin{pmatrix} \rho_{11} = 1 & \rho_{12} = \frac{\sigma_{12}}{\sqrt(\sigma_{11}\sigma_{22})} \\ \rho_{21} = \frac{\sigma_{21}}{\sqrt(\sigma_{11}\sigma_{22})} & \rho_{22} = 1 \end{pmatrix}\]Using the above equations, show that the diagonal elements of the correlation matrix must be one as defined in the above.
