Notions of information content and randomness of binary square matrix We have well established theory for measuring the information content and randomness of binary strings. Notions such as Shanon entropy and Kolmogorov-complexity were developed for binary strings.

For a binary square matrix, it is not sufficient to just convert the matrix into binary string and measure its information content or its randomness since naive unraveling of the binary matrix into binary string would lose the adjacency information in each row and in each column.

My question: What are the analogous notions for measuring the information content and randomness of binary square matrix?
You claim that unraveling the (square) matrix into a vector loses some information. However, the transformation is bijective - given the $n^2$ vector, you can easily reconstruct the $n\times n$ matrix. Information theory is oblivious to representation. If you have some random variable $X$ and some bijection defined on its domain $f$ then $H(X) = H(f(X))$. So classical information theory covers the case of matrices.