Question-6

Which of the following can be the covariance matrix of a dataset in \(\displaystyle \mathbb{R}^{2}\)?

• \(\displaystyle \begin{bmatrix} 1 & 0\\ 0 & 3 \end{bmatrix}\)

• \(\displaystyle \begin{bmatrix} -2 & 0\\ 0 & 3 \end{bmatrix}\)

• \(\displaystyle \begin{bmatrix} 1 & 2\\ 3 & 4 \end{bmatrix}\)

• \(\displaystyle \begin{bmatrix} 2 & 3.9\\ 3.9 & 8 \end{bmatrix}\)

Answer

• \(\displaystyle \begin{bmatrix} 1 & 0\\ 0 & 3 \end{bmatrix}\)

• \(\displaystyle \begin{bmatrix} -2 & 0\\ 0 & 3 \end{bmatrix}\)

• \(\displaystyle \begin{bmatrix} 1 & 2\\ 3 & 4 \end{bmatrix}\)

• \(\displaystyle \begin{bmatrix} 2 & 3.9\\ 3.9 & 8 \end{bmatrix}\)

Solution

For a matrix to be the covariance matrix of some dataset, it has to be symmetric positive semi-definite. Consider a symmetric matrix:

\[ \begin{equation*} C=\begin{bmatrix} d_{1} & a\\ a & d_{2} \end{bmatrix} \end{equation*} \]

For \(\displaystyle C\) to be positive semi-definite, its eigenvalues have to be non-negative. The characteristic polynomial is:

\[ \begin{equation*} ( d_{1} -\lambda )( d_{2} -\lambda ) -a^{2} =\lambda ^{2} -( d_{1} +d_{2}) \lambda +\left( d_{1} d_{2} -a^{2}\right) \end{equation*} \]

For the eigenvalues to be non-negative, we see that:

• \(\displaystyle d_{1} +d_{2} \geqslant 0\) and
• \(\displaystyle d_{1} d_{2} -a^{2} \geqslant 0\)

From this, we conclude that both \(\displaystyle d_{1}\) and \(\displaystyle d_{2}\) have to be non-negative, with \(\displaystyle |a|\leqslant \sqrt{d_{1} d_{2}}\). From the matrices given in the option, we see that (1) and (4) satisfy these conditions.