Real matrices (2 x 2)

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The 2 x 2 real matrices are the linear mappings of the Cartesian coordinate system into itself by the rule

$(x,y) \mapsto (x,y)\begin{pmatrix}a & c \\ b & d\end{pmatrix} = (ax + by, cx + dy).$

The set of all such real matrices is denoted by M(2,R). Two matrices p and q have a sum p + q given by matrix addition. The product matrix p q is formed from the dot product of the rows and columns of its factors through matrix multiplication. For

$q =\begin{pmatrix}a & c \\ b & d \end{pmatrix}\quad$ let $\quad q^{*} =\begin{pmatrix}d & -c \\ -b & a \end{pmatrix}$ .

Then q q * = (ad − bc) I, where I is the 2 x 2 identity matrix. The real number ad − bc is called the determinant of q. Evidently when ad − bc ≠ 0, q is an invertible matrix and q⁻¹ = q * / (ad − bc). The collection of all such invertible matrices constitutes the general linear group GL(2,R). In terms of abstract algebra, {M(2,R), +, •} forms a ring, and GL(2,R) is its group of units. M(2,R) is also a four-dimensional vector space, so it is considered an associative algebra. It is ring-isomorphic to the coquaternions, but has a different profile.

[edit] Profile

Within M(2,R), the multiples by real numbers of the identity matrix I may be considered a real line. Since every matrix lies in a commutative subring of M(2,R) that includes this real line, the whole ring can be profiled by such subrings. Toward this end one needs matrices m such that m² ∈ { −I, 0, I } to form planes P_m = {x I + ym : x, y ∈ R}, which are in fact commutative subrings.

The square of the generic matrix is

$\begin{pmatrix}aa+bc & ac+cd \\ab+bd & bc+dd \end{pmatrix}$

which is diagonal when a + d = 0. Thus we assume d = −a when looking for m to form commutative subrings. When mm = −I, then bc = −1 − aa, an equation describing an hyperbolic paraboloid in the space of parameters (a, b, c). In this case P_m is isomorphic to the field of (ordinary) complex numbers. When mm = +I, bc = +1 − aa, giving a similar surface, but now P_m is isomorphic to the ring of split-complex numbers. The case mm = 0 arises when only one of b or c is non-zero, and the commutative subring P_m is then a copy of the dual number plane.

[edit] Equi-areal mapping

Suppose X ⊂ R × R is a measurable set with area μ(X). For a given g ∈ M(2,R), let g[X] = {x g : x ∈ X}, the image of X under the linear mapping g. Then g is associated with an equi-areal mapping if μ(X) = μ(g[X]). Using the substitution rule for two variables, where the integrated function is the indicator function of X, one sees that det(g) = 1 is sufficient. Since the absolute value of the determinant appears, det(g) = −1 seems also to be sufficient. Nevertheless, in subsequent developments the concept of area is imbued with an orientation which is reversed by mappings with negative determinant, and then preservation of area includes preservation of orientation.[The study of such oriented areas, volumes, etc. involves the exterior algebra of differential forms; there orientation reversal is written

$dy\wedge dx = -( dx\wedge dy).$ ]

Thus the equi-areal mappings are identified with SL(2,R) = {g ∈ M(2,R) : det(g) = 1}, the special linear group. Given the profile above, every such g lies in a commutative subring P_m representing a type of complex plane according to the square of m. Since g g * = I, one of the following three alternatives occurs: