# Abstract Algebra/2x2 real matrices

The associative algebra of 2×2 real matrices is denoted by M(2, R). Two matrices p and q in M(2, R) have a sum p + q given by matrix addition. The product matrix p q is formed through w:matrix multiplication. For

$q={\begin{pmatrix}a&b\\c&d\end{pmatrix}},$ let
$q^{*}={\begin{pmatrix}d&-b\\-c&a\end{pmatrix}}.$ Then q q* = q*q = (adbc) I, where I is the 2×2 identity matrix. The real number ad − bc is called the determinant of q. When ad − bc ≠ 0, then q is an invertible matrix, and

$q^{-1}={\frac {q^{*}}{ad-bc}}.$ The collection of all such invertible matrices constitutes the general linear group GL(2, R). In the terms of abstract algebra, M(2, R) with the associated addition and multiplication operations 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 also an associative algebra.

The 2×2 real matrices are in one-one correspondence with the linear mappings of the two-dimensional Cartesian coordinate system into itself by the rule

$(x,\ y)\mapsto (x,\ y){\begin{pmatrix}a&c\\b&d\end{pmatrix}}=(ax+by,\ cx+dy).$ M(2,R) is where all three types of planar angle come to common expression in terms of area. M(2,R) is profiled as a 4-algebra over R, with a real line that is shared by three types of 2-algebras appearing as subalgebras of M(2,R). The division binarions, split-binarions, and dual numbers describe the three types of 2-algebras. They correspond to circular angle, hyperbolic angle, and slope respectively.

The hyperbolic angle is defined in terms of area under y=1/x. The circular angle equals the area of the corresponding sector of a circle of radius √2. Likewise, the slope equals the area of a triangle with base on a line and apex at a point √2 distance from the line.

The angles have the feature of invariance under motion, according to the type of plane. Either a rotation, a squeeze, or a shear as the case may be. Generalization of the notion of imaginary unit in M(2,R) is addressed first. It is matrix multiplication that produces the group action on a plane, so the characteristic of matrices that makes them preservers of area is addressed next.

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

To identify such m, first square the generic matrix:

${\begin{pmatrix}aa+bc&ab+bd\\ac+cd&bc+dd\end{pmatrix}}.$ 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 (abc). Such an m serves as an imaginary unit. In this case Pm 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 Pm and in this case Pm is ring isomorphism to split-binarions.

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(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.

## Equi-areal mapping

First transform one differential vector into another:

$(du,\ dv)=(dx,\ dy){\begin{pmatrix}p&q\\r&s\end{pmatrix}}=(p\ dx+r\ dy,\ \ q\ dx+s\ dy).$ Areas are measured with density $dx\wedge dy$ , a differential 2-form which involves the use of exterior algebra. The transformed density is

{\begin{aligned}du\wedge dv&=0+ps\ dx\wedge dy+qr\ dy\wedge dx+0\\&=(ps-qr)\ dx\wedge dy\\&=\det(g)\ dx\wedge dy.\end{aligned}} 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 Pm 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 a Euclidean rotation, or
• mm = I and g is a hyperbolic rotation, or
• mm = 0 and g is a shear mapping.

The preservation of area provides a common foundation for study of conformal mapping in a plane. In fact, there are three types of angles used in analysis, circular and hyperbolic angles and slope as an expression of angle in the dual number plane.

## 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 Pm described in the above profile.

First note that the invertible elements, the units, of each plane form a subgroup. The portion group that contains 1 is called the component of the identity. The polar coordinates of an element include an angle factor:

• If mm = −I, then z = ρ exp(θm) where θ is a circular angle.
• If mm = 0, then z = ρ exp(sm) or z = −ρ exp(sm) where s is a slope.
• If mm = I, then z = ρ exp(a m) or z = −p exp(a m) or
z = m ρ exp(a m) or z = −m ρ exp(a m) where a is a hyperbolic angle.

In the first case exp(θ m) = cos(θ) + m sin(θ), known as Euler's formula.

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 ${\sqrt {\rho \ \exp(am)}}={\sqrt {\rho }}\ \exp \left({\frac {1}{2}}am\right)$ 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(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 Pm must be excluded in the cases mm = 0 or mm = I.

## 2 × 2 real matrices as complex numbers

Every 2×2 real matrix can be interpreted as one of three types of (generalized) complex numbers: standard complex numbers, dual numbers, or 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 Pm. One 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

$z={\begin{pmatrix}a&b\\c&d\end{pmatrix}}.$ The complex plane Pm containing z is found as follows.

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

$z=xI+n,\quad x={\frac {a+d}{2}},\quad n=z-xI.$ Furthermore,

$n^{2}=pI$ where $p={\frac {(a-d)^{2}}{4}}+bc$ .

Now z is one of three types of complex number:

• If p < 0, then it is an ordinary complex number:
Let $q=1/{\sqrt {-p}},\quad m=qn$ . Then $m^{2}=-I,\quad z=xI+m{\sqrt {-p}}$ .
• If p = 0, then it is the dual number:
$z=xI+n$ .
• If p > 0, then z is a split-complex number:
Let $q=1/{\sqrt {p}},\quad m=qn$ . Then $m^{2}=+I,\quad z=xI+m{\sqrt {p}}$ .

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.

## Projective group

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

$[x:y]{\begin{pmatrix}a&c\\b&d\end{pmatrix}}\ =\ [ax+by:\ cx+dy].$ When cx + dy = 0, the image point is the point at infinity, otherwise
$[ax+by:\ cx+dy]\ \thicksim \left[{\frac {ax+by}{cx+dy}}:\ 1\right].$ 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 = adbc ≠ 0. Then

${\begin{pmatrix}a&c\\b&d\end{pmatrix}}\times {\begin{pmatrix}d&-c\\-b&a\end{pmatrix}}\ =\ {\begin{pmatrix}p&0\\0&p\end{pmatrix}}.$ The action of this matrix on the real projective line is

$[x:y]{\begin{pmatrix}p&0\\0&p\end{pmatrix}}\ =\ [px:py]\thicksim [x:y]$ because of projective coordinates, so that the action is that of the identity mapping on the real projective line. Therefore,
${\begin{pmatrix}a&c\\b&d\end{pmatrix}}\ {\text{and}}\ {\begin{pmatrix}d&-c\\-b&a\end{pmatrix}}$ 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.