2 × 2 real matrices

The set 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 from the dot product of the rows and columns of its factors through matrix multiplication. For

$q =\begin{pmatrix}a & b \\ c & d \end{pmatrix}, \,$

let

$\quad q^{*} =\begin{pmatrix}d & -b \\ -c & a \end{pmatrix}. \,$

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, q is an invertible matrix, and then

$q^{-1} = q^*\,/\,(ad - bc).\,$

The collection of all such invertible matrices constitutes the general linear group GL(2, R). In terms of abstract algebra, M(2, R) with the associated addition and multiplication operators 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 considered an associative algebra. It is ring-isomorphic to the coquaternions, but has a different profile.

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

$\begin{pmatrix}x \\ y\end{pmatrix} \mapsto \begin{pmatrix}a & b \\ c & d\end{pmatrix} \begin{pmatrix}x \\ y\end{pmatrix} = \begin{pmatrix}ax + by \\ cx + dy\end{pmatrix}.$

Profile

Within M(2, R), the multiples by real numbers of the identity matrix I may be considered a real line. Since every matrix lies in a commutative subring of M(2, R) that includes this real line, the whole ring can be profiled by such subrings. Toward this end one needs matrices m such that m² ∈ { −I, 0, I } to form planes P_m = {xI + ym : x, y ∈ R}, which are in fact commutative subrings.

The square of the generic matrix is

$\begin{pmatrix}aa+bc & ab+bd \\ac+cd & bc+dd \end{pmatrix} ,$

so that when a + d = 0 this square is a diagonal matrix. Thus we assume d = −a when looking for m to form commutative subrings. When mm = −I, then bc = −1 − aa, an equation describing an hyperbolic paraboloid in the space of parameters (a, b, c). In this case P_m is isomorphic to the field of (ordinary) complex numbers. When mm = +I, bc = +1 − aa, giving a similar surface, but now P_m is isomorphic to the ring of split-complex numbers. The case 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.

Equi-areal mapping

Main article: equiareal map

First transform one differential vector into another:

$\begin{pmatrix}du \\ dv \end{pmatrix} = \begin{pmatrix}p & r\\ q & s \end{pmatrix} \begin{pmatrix}dx \\ dy \end{pmatrix} = \begin{pmatrix}p\, dx + r\, dy \\ q\, dx + s\, dy\end{pmatrix}.$

Areas are measured with density $dx \wedge dy$, a differential 2-form which involves the use of exterior algebra. The transformed density is

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

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 tricotomy of planar, linear mapping in his book Linear Geometry (1965).

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 $\sqrt {\rho \ \exp (a m)} = \sqrt {\rho} \ \exp (a m /2)$ 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.

2 × 2 real matrices as complex numbers

Every 2 × 2 real matrix can be interpreted as one of three types of complex number: 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

$z = \begin{pmatrix}a & b \\ c & d \end{pmatrix}.$

We seek the complex plane P_m containing z.

As noted above, the square of a matrix 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

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

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 $m^2= - I, \quad z = x I + m \sqrt{-p}$.

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

$z=xI+n$.

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

Let $q=1/\sqrt{p}, \quad m = q n$. Then $m^2 = + I, \quad z = x I + m \sqrt{p}$.

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.

References

• 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", 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 .