1.1.1 Matrix Group Theory

Matrix groups are ubiq­ui­tous in physics and math­e­mat­ics and are heav­i­ly relied upon as rep­re­sen­ta­tions of ob­jects with sym­me­tries. Anyone who has stud­ied the Stan­dard Model in par­ti­cle physics knows that matrix groups are the foun­da­tion of the theory. From linear al­ge­bra, in­vert­ible ma­tri­ces rep­re­sent geo­met­ric mo­tions in a vector space, en­cap­su­lat­ing its geo­met­ric basis. The study of matrix groups is vast and well beyond the cur­so­ry in­tro­duc­tion found in this chap­ter, but a brief study is es­sen­tial for the reader to grasp the ma­te­r­i­al that fol­lows in the book.

De­f­i­n­i­tion (1.1) - A Matrix Group over a field F is a set of in­vert­ible ma­tri­ces in F, form­ing a group under mul­ti­pli­ca­tion, and for a matrix to be in­vert­ible, it must be square.

The field F can most com­mon­ly be rep­re­sent­ed over the real num­bers $\mathbb{R}$ or over the com­plex num­bers $\mathbb{C}$. The degree of the group $n$ is the result from the $n \times n$ size of the square ma­tri­ces that is con­sid­ered.