Real matrices (2 x 2)

The 2 x 2 real matrices are the linear mappings of the Cartesian coordinate system into itself by the rule: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: let 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 then:$q^\{-1\}\; =\; q^*\; /(ad\; -\; bc).$The collection of all such invertible matrices constitutes the general linear group GL(2,R). In terms of abstract algebra, the set of 2 by 2 real matrices and their associated addition and multiplication operators 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.

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 "m"² ∈ { −"I", 0, "I" } to form planesP_"m" = {"x I" + "ym" : "x", "y" ∈ R}, which are in fact commutative subrings.

The square of the generic matrix is

:

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.

Equi-areal mapping

First transform one differential vector into another:

:

Areas are measured with "density" $dx\; wedge\; dy$ , a differential 2-form which involves the use of exterior algebra. The transformed density is

:

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:
* "mm" = −"I" and "g" is on a circle of Euclidean rotations; or
* "mm" = "I" and "g" is on an hyperbola of squeeze mappings; or
* "mm" = 0 and "g" is on a line of shear mappings.