How to tell is a matrix is a covariance matrix? How can we know that these matrices are valid covariance matrices?
$$
       C= \begin{pmatrix}     
        1 & -1 & 2 \\
        -1 & 2 & -1 \\
        2 & -1 &  1 \\
        \end{pmatrix}
\\
       C= \begin{pmatrix}     
        4 & -1 & 1 \\
        -1 & 4 & -1 \\
        1 & -1 &  4 \\
        \end{pmatrix}
$$
I know that


*

*$C_{xy}=C_{yx}$ (is symmetric)

*$C_{xx}= \mathrm{Var}[x] \ge 0$ (the diagonal entries are all positive)


But I want to know, are there any other properties?

A square matrix is a covariance matrix of some random vector if and only if it is symmetric and positive semi-definite (see here). Positive semi-definite means that
$$
x^{T}Cx\ge0
$$
for every real vector $x$, where $x^T$ is the transpose of the vector $x$.
We have that
\begin{align*}
x^TC_1x
&=\left(\begin{array}{ccc}x_1 & x_2 & x_3\end{array}\right)\left(\begin{array}{ccc}1 & -1 & 2 \\-1 & 2 & -1 \\2 & -1 & 1\end{array}\right)\left(\begin{array}{c}x_1 \\x_2 \\x_3\end{array}\right)\\
&=\left(\begin{array}{ccc}x_1-x_2+2x_3 & -x_1+2x_2-x_3 & 2x_1-x_2+x_3\end{array}\right)\left(\begin{array}{c}x_1 \\x_2 \\x_3\end{array}\right)\\
&=x_1^2+2x_2^2+x_3^2-2x_1x_2+4x_1x_3-2x_2x_3\\
&=(x_1-x_2+x_3)^2+x_2^2+2x_1x_3.
\end{align*}
If we take $x^T=\left(\begin{array}{ccc}1 & 0 & -1\end{array}\right)$, then $x^TC_1x=-2$. Hence, $C_1$ is not a covariance matrix.
We have that
\begin{align*}
x^TC_2x
&=\left(\begin{array}{ccc}x_1 & x_2 & x_3\end{array}\right)\left(\begin{array}{ccc}4 & -1 & 1 \\-1 & 4 & -1 \\1 & -1 & 4\end{array}\right)\left(\begin{array}{c}x_1 \\x_2 \\x_3\end{array}\right)\\
&=\left(\begin{array}{ccc}4x_1-x_2+x_3 & -x_1+4x_2-x_3 & x_1-x_2+4x_3\end{array}\right)\left(\begin{array}{c}x_1 \\x_2 \\x_3\end{array}\right)\\
&=4x_1^2+4x^2+4x_3^2-2x_1x_2+2x_1x_3-2x_2x_3\\
&=3(x_1^2+x^2+x_3^2)+(x_1-x_2+x_3)^2\ge0.
\end{align*}
Hence, $C_2$ is a covariance matrix.
