(N^2) x (N^2) covariance matrix for an NxN matrix? In a given process, it is mentioned that we need to obtain a (N^2) x (N^2) covariance matrix given a NxN matrix. Since we need to have multiple sets of observations to calculate a covariance, how can this be possible?
Edit: Suppose a matrix X is generated from a multivariate normal with a mean of 0 and a given variance. The covariance structure of this generated matrix (which is somehow supposed to be (N^2) x (N^2)) is to be used as the column covariance for generating another matrix Y which will have (N^2) columns.
 A: Referring the "covariance of the covariance matrix" implies the entries in the original $N\times N$ covariance matrix $S$ must be estimates of central second moments derived from some dataset "generated from a multivariate Normal" distribution $F.$
There are many ways to construct such estimates.  They depend on what you assume about $F$ and how much data you have.  What is common to the estimates in most frequent use is that they are based on the first two empirical moments of the data.  To describe this more clearly, let the data $x_{ij}$ consist of $m$ realizations of a random variable $X$ with $N$ components $(X_1, \ldots, X_N),$ so that for each $1\le i\le m,$ $\mathrm{x}_i = (x_{i1}, \ldots, x_{iN})$ constitutes one independent realization of $X.$  The first two empirical moments are
$$\mu^{(1)}_j = \mu^{(1)}_j(\mathrm{x}_1, \ldots, \mathrm{x}_m) =\frac{1}{m} \sum_{i=1}^m x_{ij}\tag{1}$$
and
$$\mu^{(2)}_{jk} =\mu^{(2)}_{jk}(\mathrm{x}_1, \ldots, \mathrm{x}_m) = \frac{1}{m} \sum_{i=1}^m x_{ij} x_{ik}\tag{2}$$
for all $1\le j,k\le N.$  The estimated covariance between $X_j$ and $X_k$ is then given by some linear combination of these moments.  For example, the standard unbiased unweighted estimator is
$$(S^2)_{jk} = \frac{m}{m-1} \mu_{jk}^{(2)} - \frac{m}{m-1}\mu_j^{(1)}\mu_k^{(1)}.$$
For a general linear combination of such terms let's write
$$(S^2)_{jk} = \sum_{j^\prime k^\prime} \omega_{jk, j^\prime k^\prime}\,\mu_{j^\prime k^\prime}^{(2)} + \eta_{jk, j^\prime k^\prime}\,\mu_{j^\prime}^{(1)} \mu_{k^\prime}^{(1)}.\tag{3}$$
The (constant) tensors $\omega_{jkj^\prime k^\prime}$ and $\eta_{jk, j^\prime k^\prime}$ determine the estimator $S^2.$
To understand how such an estimator might vary from one sample to another, we can view the $\mathrm{x}_i$ as random variables rather than data.  Each of them has $F$ for its distribution and they are independent.  Given any two entries of an estimator $S^2(\mathrm{x}_1, \ldots, \mathrm{x}_m),$ say those with subscripts $(j,k)$ and $(j^\prime,k^\prime),$ we may thereby inquire about their covariance,
$$\operatorname{Cov}\left(S^2_{jk}, S^2_{j^\prime k^\prime}\right) = \mathbb{E}\left[\left(S^2_{jk} - \mathbb{E}\left[S^2_{jk}\right]\right) \left(S^2_{j^\prime k^\prime} - \mathbb{E}\left[S^2_{j^\prime k^\prime}\right]\right)\right].\tag{*}$$
Upon expanding the expression on the right hand side using $(1) - (3)$ above you reduce it to a linear combination of expectations of the form $E[x_{ij}\cdots x_{i^\prime j^\prime}]$ where four $x_{**}$ appear: each of these is a multivariate moment of degree $4.$ All those moments can be expressed in terms of the parameters of the distribution $F.$  When $F$ is multivariate Normal, those parameters can be taken to be its (vector) mean and its covariance matrix.
Depending on the details of the estimator $S^2$ and restrictions on those parameters, $(*)$ may simplify, but in general it's messy to write down.  I will therefore stop here, having shown what the meaning of the covariance of $S^2$ is and how it is possible to evaluate it in terms of properties of the underlying distribution $F.$
