Graph theory offers one nice illustration of the transpose (for square matrices). Here's a directed graph: ⬐---------¬ n₁ → n₂ → n₃ It can be represented

Graph theory offers one nice illustration of the transpose (for square matrices). Here's a directed graph: ⬐---------¬ n₁ → n₂ → n₃ It can be represented by this adjacency matrix: X = [ 0 1 0 ] [ 0 0 1 ] [ 1 0 0 ] See how there is a 1 for each row i and column j where there is an edge from node i to node j? Okay, now look at the transpose, Xᵀ: Xᵀ = [ 0 0 1 ] [ 1 0 0 ] [ 0 1 0 ] If we interpret this matrix as an adjacency matrix, its graph looks like this: ⬐---------¬ n₃ → n₂ → n₁ Taking the transpose of a directed graph's adjacency matrix just reverses the direction of the edges! P.S: some more cool things about adjacency matrices: Imagine that a vector represents a node or set of nodes on a graph. Like this: n₁ = [ 1 ], n₂ = [ 0 ], n₃ = [ 0 ], [ 0 ] [ 1 ] [ 0 ] [ 0 ] [ 0 ] [ 1 ] Watch what happens when we multiply X with n₁: X*n₁ = [ 0 0 1 ] [ 1 ] [ 0 ] [ 1 0 0 ] * [ 0 ] = [ 1 ] = n₂ [ 0 1 0 ] [ 0 ] [ 0 ] So the adjacency matrix is not only an index of edges, it's also a little machine that can push nodes around on a graph. You can even do it with two at a time: X*(n₁ + n₂) = [ 0 0 1 ] [ 1 ] [ 0 ] [ 1 0 0 ] * [ 1 ] = [ 1 ] = n₂ + n₃ [ 0 1 0 ] [ 0 ] [ 1 ] If you're interested in diving further into algebraic representations of graphs, check out the Leavitt Path algebra: https://arxiv.org/abs/1410.1835 https://arxiv.org/abs/1410.1835