Analogy: Matrix As a Tree

When many of us first encounter matrices, we see them as a glorified table of numbers stuffed into special square brackets.

This is analogous to looking at a tree as a collection of biological cells.

It is not incorrect; but it is perhaps a tad too granular.

One can ask, "what is the function or application of a tree or a matrix?".

For a tree, the answer may be fruits, lumber, home for birds and animals etc.

For a matrix, it may be to flip a vector, stretch it, rotate it, or some combination thereof. Here we are thinking of the tree or the matrix in terms of its relationship with other things.

One could ask what the main structural parts are.

For a tree, it may be roots, trunk, branches, and leaves.

For a matrix, it may be useful to think of it in terms of a collection of column or row vectors, and the vector spaces that they define.

One could then ask a category question: "What kind of a tree is this?"

Is it an oak tree, a fruit tree, an evergreen, a gymnosperm, etc focusing on its distinguishing properties, and the categories that it belongs to.

Similarly, one can identify different types of matrices. Is a matrix symmetric? Invertible? Triangular? Sparse? What other trees and matrices are they related to.