To further understand the shape of the multivariate normal distribution, let's return to the special case where we have p = 2 variables.

If \(ρ = 0\), there is zero correlation, and the eigenvalues turn out to be equal to the variances of the two variables. So, for example, the first eigenvalue would be equal to \(\sigma^2_1\) and the second eigenvalue would be equal to \(\sigma^2_2\)  as shown below:

\(\lambda_1 = \sigma^2_1\) and \(\lambda_2 = \sigma^2_2\)

the corresponding eigenvectors will have elements 1 and 0 for the first eigenvalue and 0 and 1 for the second eigenvalue.

\(\mathbf{e}_1 = \left(\begin{array}{c} 1\\ 0\end{array}\right)\), \(\mathbf{e}_2 = \left(\begin{array}{c} 0\\ 1\end{array}\right)\)

So, the axis of the ellipse, in this case, are parallel to the coordinate axis.

If there is zero correlation, and the variances are equal so that \(\sigma^2_1\) = \(\sigma^2_2\), then the eigenvalues will be equal to one another, and instead of an ellipse we will get a circle. In this special case, we have a so-called circular normal distribution.

If the correlation is greater than zero, then the longer axis of the ellipse will have a positive slope.

Conversely, if the correlation is less than zero, then the longer axis of the ellipse will have a negative slope.

As the correlation approaches plus or minus 1, the larger eigenvalue will approach the sum of the two variances, and the smaller eigenvalue will approach zero:

\(\lambda_1 \rightarrow \sigma^2_1 +\sigma^2_2\) and \(\lambda_2 \rightarrow 0\)

So, what is going to happen in this case is that the ellipse becomes more and more elongated as the correlation approaches one.