Abstract Algebra/2x2 real matrices

The associative algebra of 2×2 real matrices is denoted by M(2,&thinsp;R). Two matrices p and q in M(2,&thinsp;R) have a sum p + q given by matrix addition. The product matrix p&thinsp;q is formed through matrix multiplication. For
 * $$q = \begin{pmatrix} a & b \\ c & d \end{pmatrix}, $$ let
 * $$q^* = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}. $$

Then q&thinsp;q$$ = q$$&thinsp;q = (ad − bc) 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,&thinsp;R). In the terms of abstract algebra, M(2,&thinsp;R) with the associated addition and multiplication operations forms a ring, and GL(2,&thinsp;R) is its group of units. M(2,&thinsp;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 described as a pencil on a real line that is shared by three types of 2-algebras appearing as subalgebras of M(2,R). They are the division binarions, split-binarions, and dual numbers which use 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.

Pencil of planar subalgebras
In synthetic geometry, the term pencil is used for the set of lines on a given point, and axial pencil for the set of planes on a given line. Here the axis is the set of multiples of the identity matrix I by real numbers. Every matrix that is not in this set is contained in a unique planar subalgebra. These subalgebras are division-binarions, split-binarions, or dual numbers.

Given a matrix m with m2 in {I, 0, &minus;I}, there is a subalgebra
 * $$P_m = \{x I + y m : x, y \in \real \} .$$

Then Pm is a commutative subalgebra 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 subalgebras. 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 Pm is isomorphic to the division binarions, also known as 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 isomorphic 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 subalgebra Pm is then a copy of the dual number plane.

When M(2, R) is reconfigured with a change of basis, this pencil is seen in 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 $$dx \wedge dy$$, an exterior product for which $$dx \wedge dx = 0 = dy \wedge dy .$$ In linear mapping this differential 2-form is transformed:
 * $$\begin{align}

du \wedge dv &= (p \ dx + r \ dy) \wedge (q \ dx + s \ dy) \\ &= 0 + ps\ dx \wedge dy + qr\ dy \wedge dx + 0 \\ &= (ps - qr)\ dx \wedge dy \\ &= \det(g)\ dx \wedge dy. \end{align}$$

Area is preserved when the determinant is one. Thus the equi-areal mappings are identified with $$SL(2,\reals) = \{g \in M(2,\reals) : \det(g) = 1 \} ,$$ 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&#8239;g* = I, one of the following three alternatives occurs: The preservation of area provides a common foundation for study of conformal mapping in a plane. In fact, there are three species of angle used in analysis, circular and hyperbolic angle and slope as an expression of angle in the dual number plane.
 * 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.

Functions of 2 × 2 real matrices
The commutative subalgebras of M(2, R) determine the function theory; in particular the three types of subalgebras each 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 subalgebra Pm in the above pencil.

First note that the invertible elements, the units, of each plane form a topological group with one, two, or four components. The component 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 &theta; 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-binarions 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 subalgebra 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-binarions.

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 species of complex numbers
Every 2×2 real matrix can be interpreted as one of three species of (generalized ) complex number: a division binarion, a dual number, or a split-binarion. Above, the algebra of 2×2 matrices is profiled as a union of subalgebras Pm, all sharing the same real axis. One can determine to which type of subalgebra a given 2×2 matrix belongs as follows:

Consider the 2×2 matrix
 * $$z = \begin{pmatrix} a & b \\ c & d \end{pmatrix}.$$

The subalgebra Pm containing z is found by projections:

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 n one of three species of subalgebra:
 * If p < 0, then it is a division binarion:
 * 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-binarion:
 * 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-binarion plane.

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:
 * $$[x : y] \begin{pmatrix}a & c \\ b & d \end{pmatrix} \ = \ [ax + by: \ cx + dy].$$ 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 = ad – bc ≠ 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.

Autonomous differential equation
The differential equation $$f^\prime =a f$$ has solution $$f(t) = \exp(at) + C,$$ where a is a given constant and C is an arbitrary constant.

Using division binarions, the equation $$f^\prime =a f$$ may be interpreted as a tangent slope to a curve parametrized by t : for i2 = &minus; 1, the differential equation $$f^\prime =ai f$$ has solution $$f(t) = \exp(a i t) + C.$$

Similarly, for j2 = +1, the differential equation $$f^\prime =a j f$$ has solution $$f(t) = \exp(a j t) = \cosh at + j \sinh at ,$$ a branch of the unit hyperbola.

In the autonomous differential equation $$f^\prime =A f ,$$ the matrix A corresponds to a constant binarion that relates the slope of the tangent and the curve. The solution of the matrix differential equation is given by the exponential function, using this constant as a cofactor in the argument. The solution is periodic when the constant is a division binarion, and not when it is a split-binarion. Evidently the constant also determines which subalgebra of M(2,R) contains the solution curve.

Whereas linear algebra is premised on simultaneous linear equations, there is the existence theorem for solution of a system of differential equations
 * $$\frac{du_i}{dt} = \sum_{k=1}^{p} c_{ik} u_k \quad (i = 1,... p)\quad \text{where the}\quad c_{ik}$$ are continuous functions.

Here p = 2, the matrix is constant, and the solution is exhibited. However, some notational gymnastics are expected of the mathematical reader. Following tradition, the matrix is written to the left and function notation is employed. So rather than row vectors fed into the matrix in previous sections, the function is read as a column vector, which the reader must reconstruct from a binarion: