# 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 matrix multiplication. For

- let

Then *q q*^{*} = *q*^{*} *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

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

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

[edit | edit source]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 *m*^{2} in {I, 0, −I}, there is a subalgebra

Then *P*_{m} is a commutative subalgebra and M(2, **R**) = ⋃*P*_{m} where the union is over all *m* such that *m*^{2} ∈ {−I, 0, I }.

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 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 P_{m} 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 P_{m} and in this case P_{m} 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 P_{m} 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

[edit | edit source]First transform one differential vector into another:

Areas are measured with *density* , an *exterior product* for which In linear mapping this differential 2-form is transformed:

Area is preserved when the determinant is one. Thus the equi-areal mappings are identified with 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 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 species of angle used in analysis, circular and hyperbolic angle and slope as an expression of angle in the dual number plane.

## Functions of 2 × 2 real matrices

[edit | edit source]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 P_{m} 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(θ*m*) 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-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 regardless of the subalgebra 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-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 P_{m} must be excluded in the cases *mm* = 0 or *mm* = I.

## 2 × 2 real matrices as species of complex numbers

[edit | edit source]Every 2×2 real matrix can be interpreted as one of three species of (generalized^{[1]}) 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 *P*_{m}, 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

The subalgebra *P*_{m} 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 R^{4} yields

Furthermore,

- where .

Now *z* is n one of three species of subalgebra:

- If
*p*< 0, then it is a division binarion:- Let . Then .

- If
*p*= 0, then it is the dual number:- .

- If
*p*> 0, then*z*is a split-binarion:- Let . Then .

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

[edit | edit source]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*:

- When
*cx*+*dy*= 0, the image point is the*point at infinity*, otherwise

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

The action of this matrix on the real projective line is

- because of projective coordinates,

so that the action is that of the identity mapping on the real projective line. Therefore,

- 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

[edit | edit source]The differential equation has solution where *a* is a given constant and *C* is an arbitrary constant.

Using division binarions, the equation may be interpreted as a tangent slope to a curve parametrized by *t* : for i^{2} = − 1, the differential equation has solution

Similarly, for j^{2} = +1, the differential equation has solution a branch of the unit hyperbola.

In the **autonomous differential equation** 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^{[2]}

- 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:

## References

[edit | edit source]- ↑ Anthony A. Harkin & Joseph B. Harkin (2004) Geometry of Generalized Complex Numbers,
*Mathematics Magazine*77(2):118–29 - ↑ T. J. Willmore (1959)
*Introduction to Differential Geometry*, page 27, chapter 1, Appendix I, Oxford Clarendon Press

- w:Rafael Artzy (1965)
*Linear Geometry*, Chapter 2-6 Subgroups of the Plane Affine Group over the Real Field, p. 94, Addison-Wesley. - Helmut Karzel & Gunter Kist (1985) "Kinematic Algebras and their Geometries", found in
*Rings and Geometry*, R. Kaya, P. Plaumann, and K. Strambach editors, pp. 437–509, esp 449,50, D. Reidel, ISBN 90-277-2112-2 .

- Svetlana Katok (1992)
*Fuchsian groups*, pp. 113ff, University of Chicago Press ISBN 0-226-42582-7 . - Garret Sobczyk (2012). "Chapter 2: Complex and Hyperbolic Numbers".
*New Foundations in Mathematics: The Geometric Concept of Number*. Birkhäuser. ISBN 978-0-8176-8384-9.