Is this even a valid question?

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Mathematicians love to generalize existing ideas — sometimes abstracting concepts in ways that may seem detached from real-world applications. Yet, quite often, generalizations turn out to be powerful tools in science and engineering. In this article, we explore one such idea that brings together trigonometry and matrix algebra. While this appears to not be directly related to data science / AI, the mathematical techniques covered in this article are quite relevant to the field.

The broader question is simple: just as we can plug numbers into functions like sine, cosine, and exponentials, can we extend these functions to act on mathematical objects beyond real numbers — like complex numbers and matrices?

At first glance, this looks like a theoretical curiosity. However, functions of matrices play an important role in physics and engineering. They appear naturally in the study of coupled oscillations in classical mechanics and in the study of quantum mechanical systems. Functions of matrices play a very important role in the study of control theory — an area of applied mathematics used extensively in the domain of robotics.

In this article, we will mainly focus on one particularly illustrative case: applying trigonometric functions to matrices.