Exponentiating $2\times2$ and $3\times3$ Matrices Done Right

TL;DR

This paper presents explicit formulas for matrix functions like exponential, sine, cosine, hyperbolic sine, and hyperbolic cosine for 2x2 and 3x3 matrices, using only their characteristic roots, simplifying calculations especially for non-experts.

Contribution

The paper introduces formulas for matrix functions that depend solely on characteristic roots, avoiding eigenvectors and transition matrices, making computations more accessible.

Findings

Formulas for $e^A$, $ frac{d}{dt}e^{tA}$, $ ext{cosh}(A)$, $ ext{sinh}(A)$, $ ext{cos}(A)$, $ ext{sin}(A)$ for 2x2 matrices.

Explicit formulas for $e^A$ for 3x3 matrices.

Applications demonstrated through solving differential equations.

Abstract

We derive explicit formulas for calculating $e^A$, $\cosh{A}$, $\sinh{A}, \cos{A}$ and $\sin{A}$ for a given $2\times2$ matrix $A$. We also derive explicit formulas for $e^A$ for a given $3\times3$ matrix $A$. These formulas are expressed exclusively in terms of the characteristic roots of $A$ and involve neither the eigenvectors of $A$, nor the transition matrix associated with a particular canonical basis. We believe that our method has advantages (especially if applied by non-mathematicians or students) over the more conventional methods based on the choice of canonical bases. We support this point with several examples for solving first order linear systems of ordinary differential equations with constant coefficients.