What is the proof that covariance matrices are always semi-definite? Suppose that we have two different discreet signal vectors of $N^\text{th}$ dimension, namely $\mathbf{x}[i]$ and $\mathbf{y}[i]$, each one having a total of $M$ set of samples/vectors.
$\mathbf{x}[m] = [x_{m,1} \,\,\,\,\, x_{m,2} \,\,\,\,\, x_{m,3} \,\,\,\,\, ... \,\,\,\,\, x_{m,N}]^\text{T}; \,\,\,\,\,\,\, 1 \leq m \leq M$
$\mathbf{y}[m] = [y_{m,1} \,\,\,\,\, y_{m,2} \,\,\,\,\, y_{m,3} \,\,\,\,\, ... \,\,\,\,\, y_{m,N}]^\text{T}; \,\,\,\,\,\,\,\,\, 1 \leq m \leq M$
And, I build up a covariance matrix in-between these signals.
$\{C\}_{ij} = E\left\{(\mathbf{x}[i] - \bar{\mathbf{x}}[i])^\text{T}(\mathbf{y}[j] - \bar{\mathbf{y}}[j])\right\}; \,\,\,\,\,\,\,\,\,\,\,\, 1 \leq i,j \leq M $
Where, $E\{\}$ is the "expected value" operator.
What is the proof that, for all arbitrary values of $\mathbf{x}$ and $\mathbf{y}$ vector sets, the covariance matrix $C$ is always semi-definite ($C \succeq0$) (i.e.; not negative definte; all of its eigenvalues are non-negative)?

Covariance matrix C is calculated by the formula,
$$
\mathbf{C} \triangleq E\{(\mathbf{x}-\bar{\mathbf{x}})(\mathbf{x}-\bar{\mathbf{x}})^T\}.
$$
Where are going to use the definition of positive semi-definite matrix which says:

A real square matrix $\mathbf{A}$ is positive semi-definite if and only if
$\mathbf{b}^T\mathbf{A}\mathbf{b}\succeq0$
is true for arbitrary real column vector $\mathbf{b}$ in appropriate size.

For an arbitrary real vector u, we can write,
$$
\begin{array}{rcl}
\mathbf{u}^T\mathbf{C}\mathbf{u} & = & \mathbf{u}^TE\{(\mathbf{x}-\bar{\mathbf{x}})(\mathbf{x}-\bar{\mathbf{x}})^T\}\mathbf{u} \\
& = & E\{\mathbf{u}^T(\mathbf{x}-\bar{\mathbf{x}})(\mathbf{x}-\bar{\mathbf{x}})^T\mathbf{u}\} \\
& = & E\{s^2\} \\
& = & \sigma_s^2. \\
\end{array}
$$
Where $\sigma_s$ is the variance of the zero-mean scalar random variable $s$, that is,
$$
s = \mathbf{u}^T(\mathbf{x}-\bar{\mathbf{x}}) = (\mathbf{x}-\bar{\mathbf{x}})^T\mathbf{u}.
$$
Square of any real number is equal to or greater than zero.
$$
\sigma_s^2 \ge 0
$$
Thus,
$$
\mathbf{u}^T\mathbf{C}\mathbf{u} = \sigma_s^2 \ge 0.
$$
Which implies that covariance matrix of any real random vector is always positive semi-definite.
