Within M, the multiples by real numbers of the identity matrix may be considered a real line. This real line is the place where all commutative subrings come together: Let Pm = where m2 ∈ . Then Pm is a commutative subring and M = ⋃Pm where the union is over all m such that m2 ∈ . To identify such m, first square the generic matrix: When a + d = 0 this square is a diagonal matrix. Thus one assumes d = −a when looking for m to form commutative subrings. When mm = −, then bc = −1 − aa, an equation describing a hyperbolic paraboloid in the space of parameters. Such an m serves as an imaginary unit. In this case Pm is isomorphic to the field of complex numbers. When mm = +, m is an involutory matrix. Then bc = +1 − aa, also giving a hyperbolic paraboloid. If a matrix is an idempotent matrix, it must lie in such a Pm and in this case Pm is isomorphic to the ring of split-complex numbers. The case of a nilpotent matrix, mm = 0, arises when only one of b or c is non-zero, and the commutative subring Pm is then a copy of the dual number plane. When M is reconfigured with a change of basis, this profile changes to the profile of split-quaternions where the sets of square roots of and − take a symmetrical shape as hyperboloids.
Equi-areal mapping
First transform one differential vector into another: Areas are measured with density, a differential 2-form which involves the use of exterior algebra. The transformed density is Thus the equi-areal mappings are identified with SL = , the special linear group. Given the profile above, every such g lies in a commutative subring Pm representing a type of complex plane according to the square of m. Since g g = , one of the following three alternatives occurs:
mm = − and g is on a circle of Euclidean rotations; or
mm = and g is on an hyperbola of squeeze mappings; or
The commutative subrings of M determine the function theory; in particular the three types of subplanes have their own algebraic structures which set the value of algebraic expressions. Consideration of the square root function and the logarithm function serves to illustrate the constraints implied by the special properties of each type of subplane Pm described in the above profile. The concept of identity component of the group of units of Pm leads to the polar decomposition of elements of the group of units:
If mm = −, then z = ρ exp.
If mm = 0, then z = ρ exp or z = −ρ exp.
If mm = , then z = ρ exp or z = −ρ exp or z = m ρ exp or z = −m ρ exp.
In the first case exp = cos + m sin. In the case of the dual numbers exp = 1 + s m. Finally, in the case of split complex numbers there are four components in the group of units. The identity component is parameterized by ρ and exp = cosh + m sinh. Now regardless of the subplane Pm, but the argument of the function must be taken from the identity component of its group of units. Half the plane is lost in the case of the dual number structure; three-quarters of the plane must be excluded in the case of the split-complex number structure. Similarly, if ρ exp is an element of the identity component of the group of units of a plane associated with matrix m, then the logarithm function results in a value log ρ + a m. The domain of the logarithm function suffers the same constraints as does the square root function described above: half or three-quarters of Pm must be excluded in the cases mm = 0 or mm = . Further function theory can be seen in the article complex functions for the C structure, or in the article motor variable for the split-complex structure.
2 × 2 real matrices as complex numbers
Every real matrix can be interpreted as one of three types of complex numbers: standard complex numbers, dual numbers, and split-complex numbers. Above, the algebra of matrices is profiled as a union of complex planes, all sharing the same real axis. These planes are presented as commutative subrings Pm. We can determine to which complex plane a given matrix belongs as follows and classify which kind of complex number that plane represents. Consider the matrix We seek the complex plane Pm containing z. As noted above, the square of the matrix z is diagonal when a + d = 0. The matrix z must be expressed as the sum of a multiple of the identity matrix and a matrix in the hyperplanea + d = 0. Projecting z alternately onto these subspaces of R4 yields Furthermore, Now z is one of three types of complex number:
If p < 0, then it is an ordinary complex number:
: Let. Then.
If p = 0, then it is the dual number:
:.
If p > 0, then z is a split-complex number:
: Let. Then.
Similarly, a matrix can also be expressed in polar coordinates with the caveat that there are two connected components of the group of units in the dual number plane, and four components in the split-complex number plane.
Projective group
A given 2 × 2 real matrix with ad ≠ bc acts on projective coordinates of the real projective lineP as a linear fractional transformation: Rather than acting on the plane as in the section above, a matrix acts on the projective line P, and all proportional matrices act the same way. Let p = ad – bc ≠ 0. Then The action of this matrix on the real projective line is The projective group starts with the group of units GL of M, and then relates two elements if they are proportional, since proportional actions on P are identical:
