Covariance Matrix of two random variables

Question

Can two random variables $X$ and $Y$ have the covariance matrix

$\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}? $

The matrix is positive definite if $xy<0$. Does this imply that it could be the convariance matrix for some $X$ and $Y$?

Answer 1

The eigenvalues of the matrix $$ \begin{pmatrix} 0&-1\\ -1&0 \end{pmatrix} $$ are $-1$ and $1$. The eigenvalues of a positive semidefinite matrix are non-negative. Hence, this matrix is not positive semidefinite and it is not a covariance matrix.

Answer 2

In order to show that the matrix $A=\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}$ can't be a covariance matrix, we need to show that it's not a PSD matrix, i.e., it's sufficient to show an example vector ($\exists \mathbb{x} \in \mathbb{R^2}$), such that $x^TAx<0$.

Let $x=[1,1]^T$, then we have

$x^TAx=\begin{pmatrix} 1 & 1 \end{pmatrix}. \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}.\begin{pmatrix} 1 \\ 1 \end{pmatrix}=\begin{pmatrix} -1 & -1 \end{pmatrix}.\begin{pmatrix} 1 \\ 1 \end{pmatrix}=-2 < 0$.

Hence, $A$ is not a PSD matrix and can't be a covariance matrix.