for random variable X = (x1,x2,...,xn)T, I understand the entries of the covariance matrix would just be the covariance of xi and xj, but how to find the eigenvalues and eigenvectors after that, and how does that turn into the eigendecomposition of the covariance matrix?

For a random variable $X = (x_1,x_2,\ldots,x_n)^T$, I understand that the entries of the covariance matrix would just be the covariance of $x_i$ and $x_j$, but how do I find the eigenvalues and eigenvectors after that, and how does that turn into the eigendecomposition of the covariance matrix?