The difference between a matrix valued random variable and an $n \times p$ matrix of data

Question

So I am totally new to the field of random matrices, but I was not sure about how they are applied. According to Wikipedia, a random matrix is "a matrix-valued random variable—that is, a matrix some or all of whose elements are random variables." However, I don't understand where I would use a random matrix in a statistical model versus a random matrix. In other words, in multivariate statistical analysis we usually have an $n \times p$ data matrix, where given an observation $X_i$, we collect various $p$ attributes of this observation, so $X_{i,1}, . . ., X_{i,p}$. So why can't we represent the elements of a random matrix as an $n \times p$ matrix like we use in multivariate statistics? Sorry if this question is really naive.

Answer 1

What you have described for $X$ is just the matrix of predictors. A random matrix is quite different. Lets start with a perhaps more familiar example, a bivariate gaussian random variable. This "variable" is a vector valued random variable $X=(X_1,X_2)$, where the two components of $X$ can co-vary with each other (i.e., to get an "oblong" shaped gaussian density).

Therefore, a "vector valued" random variable is associated with a joint distribution, $p(x,y,z...)$ of a set of scalar-valued random variables.

Generalizing this further, if each component is itself a vector, then you get a "matrix valued" random variable. This may be useful when you are describing the joint distribution of a set of vector-valued quantities. For example, let $X=(\nabla P,\nabla T, \nabla v)$, where $P$ is pressure,$T$ is temperature, and $v$ is the wind velocity for a given point. If these quantities are not independent of one another, then it makes sense to treat them as part of a single, matrix valued random variable $X$