Let's say I have a matrix where some of the components are random variables. From Wikipedia, the expectation of the matrix is simply the expectation of the individual components. But, beyond that, what is the possible meaning of getting 'an expectation of a matrix'?
For example, if we toss a fair die overnight and record the number that appears on top as the die lands, the average of these numbers is definitely 3.5. This is how we picture the expectation of a random variable.
How could we extend this to expectation of matrices? And what could the possible implications of the eigenvalues and stability of the matrix?
Any insights?
There is nothing much special about it; not more than the expectation of a multivariate random variable, which is usually represented as a vector (or tuple)... or a $1\times n$ (or $n\times 1$) matrix, ${\bf X}=(X_1, X_2 \cdots X_n)$. The expectation is an element-wise operation, and the expectation of the matrix (or vector) is just the matrix (or vector) consisting of the expectation of each element - and the same goes for how you "picture" it (for example, in relation with the law of large numbers).
In regard with other values that are function of the random matrix (say, determinant, trace, eigenvalues...) you cannot say something in general, unless the function is linear. Then, for example, one could say that the expected trace of a random matrix is the trace of the expectation. But that cannot be said a priori for the determinant, or the eigenvalues.