# 2 \(\times\) 2 Matrices: Manifolds, Realizations, Applications
## 1 Introduction: Linear Transformations  
We are familiar with the use of vectors and matrices, both as a convenient shorthand to write related sets of values and equations, and as a transparent way of expressing physical quantities whose description depends on position, space orientation, or time, through linear transformations. The study of matrices _per_ is generally credited to [PERSON] [1] and several of his contemporaries and followers, including the encyclopedic work of [PERSON] [2] on determinants. The theory of finite groups, developed in the second half of the 19 th century, particularly resulting from [PERSON]'s lemma [3], binds the number of group elements with the number of entries of all their matrix representations. The golden age for group theory, turning on continuous groups, was started by [PERSON] with his doctoral thesis at the University of Christiania (Oslo) [4]; the continuous groups were finally classified by [PERSON] [5] into four classical families: Unitary, symplectic, and odd and even-dimensional orthogonal groups--plus the five exceptional groups. The 20 th century witnessed [PERSON]'s matrix version of quantum mechanics, the recognition of spin as a fundamental non-classical property of matter, [PERSON]'s development of angular momentum couplings, [PERSON]'s relativistic electron equation, and more recently, linear optics and canonical transforms [6, 7, 8]. Most of the applications in quantum angular momentum mechanics involve large matrices that nevertheless depend on only a few 'active' parameters, which are actually represented by \(2\times 2\) (or special \(4\times 4\)) matrices. By the mid-20 th century, the mathematical theory of Hilbert spaces and Gel'fand triplets [9] allowed the consideration of'matrices' with a continuum of rows and columns, which are properly called _integral kernels_. To underscore these constructs and methods, we consider sufficient to work with the lowest \(2\times 2\) case, pointing out some features which are generic in higher dimensions. For plane geometric optics or mechanics, the manifold of canonical transforms is the group of \(2\times 2\) real matrices. The fundamental property of the set of square matrices is its closure under _multiplication_. In the \(2\times 2\) case, the product of \(\mathbf{M}_{1}=\left( \begin{smallmatrix}a_{1}&b_{1}\\ c_{1}&d_{1}\end{smallmatrix} \right)\) and \(\mathbf{M}_{2}=\left( \begin{smallmatrix}a_{2}&b_{2}\\ c_{2}&d_{2}\end{smallmatrix} \right)\), is:  
\[\left( \begin{smallmatrix}a_{1}&b_{1}\\ c_{1}&d_{1}\end{smallmatrix} \right)\left( \begin{smallmatrix}a_{2}&b_{2}\\ c_{2}&d_{2}\end{smallmatrix} \right)=\left( \begin{smallmatrix}a_{1}a_{2}+b_{1} c_{2}&a_{1}b_{2}+b_{1}d_{2}\\ c_{1}a_{2}+d_{1}c_{2}&c_{1}b_{2}+d_{1}d_{2}\end{smallmatrix} \right), \tag{1}\]  
where the entries \(a_{1}\),..., \(d_{2}\) are numbers in some field: Real \(\mathbf{R}\) or complex \(\mathbf{C}\) (other fields, such as quaternions or quadratic extensions \(x+y\surd 2\), etc., are seldom considered). This product is not generally commutative (\(\mathbf{M}_{1}\mathbf{M}_{2}\
eq\mathbf{M}_{2}\mathbf{M}_{1}\)), but it is associative: \((\mathbf{M}_{1}\mathbf{M}_{2})\mathbf{M}_{3}=\mathbf{M}_{1}(\mathbf{M}_{2} \mathbf{M}_{3})\), and it has a single unit element \(\mathbf{1}=\left( \begin{smallmatrix}1&0\\ 0&1\end{smallmatrix} \right)\) that satisfies \(\mathbf{M}\,\mathbf{1}=\mathbf{M}=\mathbf{1}\,\mathbf{M}\) for all \(\mathbf{M}\). These are three of the four axioms these matrices need to fit in a mathematical structure called a group. The fourth axiom is the existence of a unique inverse for every matrix \(\mathbf{M}=\left( \begin{smallmatrix}a&b\\ c&d\end{smallmatrix} \right)\), which is given by:  
\[\mathbf{M}^{-1}=\frac{1}{ad-bc}\left( \begin{smallmatrix}d&-b\\ -c&a\end{smallmatrix} \right),\qquad\text{provided}\;\det\mathbf{M}=ad-bc\
eq 0, \tag{2}\]  
and such that \(\mathbf{M}\,\mathbf{M}^{-1}=\mathbf{1}=\mathbf{M}^{-1}\mathbf{M}\). This last axiom restricts the matrices belonging to a group to have a non-zero determinant.