I have always been fascinated by matrices. Indeed, my parents told me that when I first learned to write I would spend hours writing out pages of zeros and ones in an array. It is no wonder that matrices of 0s and 1s have played a big role in my mathematical adventures.

According to the Miriam-Webster dictionary, mesmerize means “to interest or amaze (someone) so much that nothing else is seen or noticed.” Given this, it is fair to say that Matrices Mesmerize Me.

In his delightful book Mathematics for Human Flourishing, Frances Su advocates for sharing the joy of mathematics with others. Motivated by this, I am starting this blog that will highlight some of my mathematical joys. The primary focus will be on matrices (individual ones, families, results about) that have mesmerized me.

I imagine most of the posts will be informal, light, and hopefully enjoyable reading. Some may concern deeper concepts and theory, and for those references will be provided.

For this initial post, I’ll simply describe the 3 by 3 matrix in the logo for the blog. Replacing the three brown hexagons by ones, and the six gold hexagons by zeros results in the 3 by 3 matrix.

This is a permutation matrix; that is, a matrix of 0s and 1s with exactly one 1 in each column and each row.

Now form the vector between any two ones in the matrix. We get the vectors [-1, 1], [-1,2] and [-2,1] (or their negatives). No two of these three vectors point in the same direction!

An n by n permutation matrix with the property that the n(n-1)/2 vectors between pairs of 1s are in distinct directions is called a Costas array. So the matrix in the logo is a Costas array.

In a forthcoming post, I’ll describe how Costas arrays arise in applications of sonar and radar; and how constructions Costas array motivate interesting (still open) problems about finite fields.