Imagine, if you will, a polynomial of degree 2: ax^2 + bx + c. Now, replace a, b and c with square matrices A, B and C to form what is known as a quadratic matrix polynomial that looks like Ax^2 + Bx + C. In this poster, I will take you on a journey to construct a beautiful set called the numerical range of a matrix polynomial. We will plot points from some examples to see why they are beautiful on a graphical level, making note of the symmetry and complexity of the patterns formed. In addition, we will observe other properties of the numerical range that will allow us a glimpse of the intricate beauty of the math behind it and show us that they are highly non-trivial to describe completely. For those with more pragmatic minds, before our journey is over, we will have seen an application of matrix polynomials to an engineering problem. The hope by the end of this adventure is to open your eyes to a branch of mathematics that you may have never heard about and to encourage you to find beauty in the complexity of mathematical constructs like the numerical range.

A Beautiful Set and Matrix Polynomials: Exploring the Geometry of The Numerical Range