1. Profile

Within M(2, R), the multiples by real numbers of the identity matrix I may be considered a real line. This real line is the place where all commutative subrings come together:

Let P_m = {xI + ym : x, y ∈ R} where m² ∈ {−I, 0, I }. Then P_m is a commutative subring and M(2, R) = ⋃P_m  where the union is over all m such that m² ∈ {−I, 0, I }.

To identify such m, first square the generic matrix:

[math]\displaystyle{ \begin{pmatrix}aa + bc & ab + bd \\ ac + cd & bc + dd \end{pmatrix}. }[/math]

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 = −I, then bc = −1 − aa, an equation describing a hyperbolic paraboloid in the space of parameters (a, b, c). Such an m serves as an imaginary unit. In this case P_m is isomorphic to the field of (ordinary) complex numbers.

When mm = +I, 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 P_m and in this case P_m 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 P_m is then a copy of the dual number plane.

When M(2, R) is reconfigured with a change of basis, this profile changes to the profile of split-quaternions where the sets of square roots of I and −I take a symmetrical shape as hyperboloids.

2. Equi-areal Mapping

First transform one differential vector into another:

[math]\displaystyle{ \begin{pmatrix}du \\ dv \end{pmatrix} = \begin{pmatrix}p & r\\ q & s \end{pmatrix} \begin{pmatrix}dx \\ dy \end{pmatrix} = \begin{pmatrix}p\, dx + r\, dy \\ q\, dx + s\, dy\end{pmatrix}. }[/math]

Areas are measured with density [math]\displaystyle{ dx \wedge dy }[/math], a differential 2-form which involves the use of exterior algebra. The transformed density is

[math]\displaystyle{ \begin{align} du \wedge dv &= 0 + ps\ dx \wedge dy + qr\ dy \wedge dx + 0 \\ &= (ps - qr)\ dx \wedge dy \\ &= \det(g)\ dx \wedge dy. \end{align} }[/math]

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.

Writing about planar affine mapping, Rafael Artzy made a similar trichotomy of planar, linear mapping in his book Linear Geometry (1965).

3. Functions of 2 × 2 Real Matrices

The commutative subrings of M(2, R) 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 P_m described in the above profile. The concept of identity component of the group of units of P_m leads to the polar decomposition of elements of the group of units:

• If mm = −I, then z = ρ exp(θm).
• If mm = 0, then z = ρ exp(s m) or z = −ρ exp(s m).
• If mm =  I, then z = ρ exp(a m) or z = −ρ exp(a m) or z = m ρ exp(a m) or z = −m ρ exp(a m).

In the first case exp(θ m) = cos(θ) + m sin(θ). In the case of the dual numbers exp(s m) = 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(a m) = cosh(a) + m sinh(a).

Now [math]\displaystyle{ \sqrt{\rho\ \exp(am)} = \sqrt{\rho}\ \exp\left(\frac{1}{2}am\right) }[/math] regardless of the subplane P_m, 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(a m) is an element of the identity component of the group of units of a plane associated with 2×2 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 P_m must be excluded in the cases mm = 0 or mm = I.

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.

4. 2 × 2 Real Matrices as Complex Numbers

Every 2×2 real matrix can be interpreted as one of three types of (generalized^[1]) complex numbers: standard complex numbers, dual numbers, and split-complex numbers. Above, the algebra of 2×2 matrices is profiled as a union of complex planes, all sharing the same real axis. These planes are presented as commutative subrings P_m. We can determine to which complex plane a given 2×2 matrix belongs as follows and classify which kind of complex number that plane represents.

Consider the 2×2 matrix

[math]\displaystyle{ z = \begin{pmatrix} a & b \\ c & d \end{pmatrix}. }[/math]

We seek the complex plane P_m 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 I and a matrix in the hyperplane a + d = 0. Projecting z alternately onto these subspaces of R⁴ yields

[math]\displaystyle{ z = xI + n ,\quad x = \frac{a + d}{2}, \quad n = z - xI. }[/math]

Furthermore,

[math]\displaystyle{ n^2 = pI }[/math] where [math]\displaystyle{ p = \frac{(a - d)^2}{4} + bc }[/math].

Now z is one of three types of complex number:

• If p < 0, then it is an ordinary complex number:

  Let [math]\displaystyle{ q = 1/\sqrt{-p}, \quad m = qn }[/math]. Then [math]\displaystyle{ m^2 = -I, \quad z = xI + m\sqrt{-p} }[/math].

• If p = 0, then it is the dual number:

  [math]\displaystyle{ z = xI + n }[/math].

• If p > 0, then z is a split-complex number:

  Let [math]\displaystyle{ q = 1/\sqrt{p}, \quad m = qn }[/math]. Then [math]\displaystyle{ m^2 = +I, \quad z = xI + m\sqrt{p} }[/math].

Similarly, a 2×2 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.

5. Projective Group

A given 2 × 2 real matrix with ad ≠ bc acts on projective coordinates [x : y] of the real projective line P(R) as a linear fractional transformation:

[math]\displaystyle{ [x : y] \begin{pmatrix}a & c \\ b & d \end{pmatrix} \ = \ [ax + by: \ cx + dy]. }[/math] When cx + dy = 0, the image point is the point at infinity, otherwise

[math]\displaystyle{ [ax + by:\ cx + dy] \ \thicksim \left[\frac{ax + by} {cx + dy} : \ 1\right] . }[/math]

Rather than acting on the plane as in the section above, a matrix acts on the projective line P(R), and all proportional matrices act the same way.

Let p = ad – bc ≠ 0. Then

[math]\displaystyle{ \begin{pmatrix}a & c \\ b & d \end{pmatrix} \times \begin{pmatrix} d & -c \\ -b & a \end{pmatrix} \ = \ \begin{pmatrix} p & 0 \\ 0 & p \end{pmatrix} . }[/math]

The action of this matrix on the real projective line is

[math]\displaystyle{ [x : y] \begin{pmatrix}p & 0 \\ 0 & p \end{pmatrix} \ = \ [px : py] \thicksim [x : y] }[/math] because of projective coordinates, so that the action is that of the identity mapping on the real projective line. Therefore,

[math]\displaystyle{ \begin{pmatrix}a & c \\ b & d \end{pmatrix} \ \text{and} \ \begin{pmatrix} d & -c \\ -b & a \end{pmatrix} }[/math] act as multiplicative inverses.

The projective group starts with the group of units GL(2,R) of M(2,R), and then relates two elements if they are proportional, since proportional actions on P(R) are identical:

PGL(2,R) = GL(2,R)/~ where ~ relates proportional matrices. Every element of the projective linear group PGL(2,R) is an equivalence class under ~ of proportional 2 × 2 real matrices.