Assume I have a matrix $X = [X_1,X_2, \dots, X_N ]$ where each $X_i$ is a column vector of some data of length $M$.
Compute the correlation matrix that contains all of the pairwise correlations among the columns of the matrix $X$.
I don't understand what this is asking. How do I take this set of columns, compute their pairwise correlations, and combine them into one matrix?
This means that the coefficient $(i,j)$ of the covariance matrix is $\operatorname{Cov}\left(X_i,X_j\right)$, $1\leqslant i,j\leqslant N$.