Both covariance matrices and Hessians are more naturally thought of as tensors, not matrices (and therefore not linear transformations). That is, they take in two vectors as input and produce a single real number as output.
As for graph adjacency matrix, this can actually be thought of as a linear transformation on the vector space where the basis vectors correspond to nodes in the graph. Linear combinations of these basis vectors correspond to probability distributions over the graph (if properly normalized).
2D images... Yes, these cannot really be interpreted as linear transformations. But I'd say these aren't really matrices in the mathematical sense.
I politely disagree about covariance and Hessians. I can squint and say that the Hessian provides a change in gradient when multiplied by a delta vector. Similarly for covariance... Or you could look at it as one half of the dot product for a Bhattacharyya distance, which is just a product of three matrices (row vector, square matrix, col vector). No need for tensors yet.
That is unless you decide to squint hard enough to see everything as tensors! :-)
> Linear algebra is the branch of mathematics concerning linear equations, linear functions, and their representations in vector spaces through matrices. [0]
The Venn Diagram of Linear Algebra and Matrices definitely has a lot of non-overlap, of which your list covers some. This article should be renamed to be about matrices and not linear algebra, because it's not.
This can be useful in, say, exotic options trading - a natural unit of measurement is how many ‘vols’ an underlier has moved, e.g. a 10-vol move is very large.
And vice versa, btw - take iid RV with unit variance, hit them with the Cholesky decomposition, and you have the desired covariance. Used all over Monte Carlo and finance and so on.
Let's leave the word 'basis' out, since the column vectors may well be linearly dependent.