Matrices and graphs

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Matrices and graphs

Understanding math will make you a better engineer.

So, I am writing the best and most comprehensive book about it.

The single most undervalued fact of linear algebra: matrices are graphs, and graphs are matrices.

Encoding matrices as graphs is a cheat code, making complex behavior simple to study.

Let me show you how!

A matrix and its graph

If you looked at the example above, you probably figured out the rule: each row is a node, and each element represents a directed and weighted edge. The element in the i\textstyle i-th row and j\textstyle j-th column corresponds to an edge going from i\textstyle i to j\textstyle j.

Why is the directed graph representation beneficial for us?

For one, the powers of the matrix correspond to walks in the graph. Take a look at the elements of the square matrix. All possible 2-step walks are accounted for in the sum defining the elements of A2A^2.

Powers of a matrix and its interpretation with graphs

(If the directed graph represents the states of a Markov chain, the square of its transition probability matrix essentially shows the probability of the chain having some state after two steps.)

There is much more to this connection. For instance, it gives us a deep insight into the structure of nonnegative matrices. To see what graphs show about matrices, let's talk about the concept of strongly connected components.

A directed graph is strongly connected if every node can be reached from every other node.

Below, you can see two examples where this holds and doesn't hold.

Strongly connectedness

Matrices that correspond to strongly connected graphs are called irreducible. All other nonnegative matrices are called reducible. Soon, we'll see why.

(For simplicity, I assumed each edge to have unit weight, but each weight can be an arbitrary nonnegative number.)

Irreducible matrices

Back to the general case!

Even though not all directed graphs are strongly connected, we can partition the nodes into strongly connected components.

Strongly connected components

Let's label the nodes of this graph and construct the corresponding matrix! (For simplicity, assume that all edges have unit weight.)

Strongly connected components and their matrix

Do you notice a pattern?

The corresponding matrix of our graph can be reduced to a simpler form! Its diagonal comprises blocks whose graphs are strongly connected. (That is, the blocks are irreducible.) Furthermore, the block below the diagonal is zero.

Frobenius normal form example

In general, this block-matrix structure is called the Frobenius normal form.

Frobenius normal form

Let's reverse the question: can we transform an arbitrary nonnegative matrix into the Frobenius normal form?

Yes, and with the help of directed graphs, this is much easier to show than purely using algebra. We have already seen the proof:

  1. construct the corresponding directed graph,
  2. find its strongly connected components,
  3. and renumber its nodes such that the components' nodes form blocks among the integers.

Without going into the details, renumbering the nodes is equivalent to reordering the rows and the columns of our original matrix, resulting in the Frobenius normal form.

Frobenius normal form

This is just the tip of the iceberg. For example, with the help of matrices, we can define the eigenvalues of graphs!

Utilizing the relation between matrices and graphs has been extremely profitable for both graph theory and linear algebra.

Having a deep understanding of math will make you a better engineer.

I want to help you with this, so I am writing a comprehensive book that takes you from high school math to the advanced stuff.
Join me on this journey and let's do this together!