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In [[mathematics]], the [[set (mathematics)|set]] of '''[[2 (number)|2]]&nbsp;&times;&nbsp;2 [[real number|real]] [[matrix (mathematics)|matrices]]''' is denoted by M(2,&thinsp;'''R'''). Two matrices ''p'' and ''q'' in M(2,&thinsp;'''R''') have a sum ''p''&nbsp;+&nbsp;''q'' given by [[matrix addition]]. The product matrix {{nowrap|''p&thinsp;q''}} is formed from the [[dot product]] of the rows and columns of its factors through [[matrix multiplication]]. For

: <math>q =\begin{pmatrix}a & b \\ c & d \end{pmatrix}, \,</math>

let

: <math>\quad q^{*} =\begin{pmatrix}d & -b \\ -c & a \end{pmatrix}. \,</math>

Then ''q&thinsp;q''&thinsp;* = ''q''&thinsp;*''q''&thinsp; = (''ad'' &minus; ''bc'') ''I'', where  ''I'' is the 2&nbsp;&times;&nbsp;2 identity matrix. The real number  ''ad''&nbsp;&minus;&nbsp;''bc''  is called the [[determinant]] of ''q''. When  ''ad''&nbsp;&minus;&nbsp;''bc''&nbsp;≠&nbsp;0, ''q'' is an [[invertible matrix]], and then

:<math>q^{-1} = q^*\,/\,(ad - bc).\,</math>

The collection of all such invertible matrices constitutes the [[general linear group]] GL(2,&thinsp;'''R'''). In terms of [[abstract algebra]], M(2,&thinsp;'''R''') with the associated addition and multiplication operators forms a [[ring (mathematics)|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 considered an [[associative algebra]]. It is ring-isomorphic to the [[coquaternion]]s, but has a different profile.

The '''2&nbsp;&times;&nbsp;2 real matrices''' are in [[one-one correspondence]] with the [[linear mapping]]s of the [[Cartesian coordinate system#Cartesian coordinates in two dimensions|two-dimensional Cartesian coordinate system]] into itself by the rule

:<math>\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}.</math>

==Profile==
Within M(2,&nbsp;'''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 [[subring]]s come together:

Let ''P''<sub>''m''</sub>&nbsp;=&nbsp;{''xI''&nbsp;+&nbsp;''ym''&nbsp;:&nbsp;''x'',&nbsp;''y''&nbsp;∈&nbsp;'''R'''} where  ''m''<sup>2</sup>&nbsp;∈&nbsp;{&nbsp;&minus;''I'',&nbsp;0,&nbsp;''I''&nbsp;}. Then ''P''<sub>''m''</sub> is a commutative subring and M(2,&nbsp;'''R''') = ∪''P''<sub>''m''</sub> &nbsp; where the union is over all ''m'' such that ''m''<sup>2</sup>&nbsp;∈&nbsp;{&nbsp;&minus;''I'',&nbsp;0,&nbsp;''I''&nbsp;}. 

To identify such ''m'', first square the generic matrix:
:<math>\begin{pmatrix}aa+bc & ab+bd \\ac+cd & bc+dd \end{pmatrix} .</math>
When  ''a'' + ''d'' = 0 this square is a [[diagonal matrix]].
Thus we assume ''d''&nbsp;=&nbsp;&minus;''a'' when looking for ''m'' to form commutative subrings. When ''mm''&nbsp;=&nbsp;&minus;''I'', then ''bc''&nbsp;=&nbsp;&minus;1&nbsp;&minus;&nbsp;''aa'', an equation describing an [[hyperbolic paraboloid]] in the space of parameters (''a'',&nbsp;''b'',&nbsp;''c''). In this case P<sub>''m''</sub> is isomorphic to the field of (ordinary) [[complex number]]s. When ''mm''&nbsp;=&nbsp;+''I'', ''bc''&nbsp;=&nbsp;+1&nbsp;&minus;&nbsp;''aa'', giving a similar surface, but now  P<sub>''m''</sub> is isomorphic to the ring of [[split-complex number]]s. The case ''mm''&nbsp;=&nbsp;0 arises when only one of  ''b'' or ''c'' is non-zero, and the commutative subring P<sub>''m''</sub> 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 [[coquaternion#Profile|profile of split-quaternions]] where the sets of square roots of ''I'' and &minus;''I'' take a symmetrical shape as [[hyperboloid]]s.

==Equi-areal mapping==
<!-- This section is linked from [[Area]] -->
{{main|equiareal map}}
First transform one differential vector into another:
:<math>
\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>

[[Area]]s are measured with ''density'' <math>dx \wedge dy</math>, a [[differential form|differential 2-form]] which involves the use of [[exterior algebra]]. The transformed density is

:<math>
\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 
[[SL2(R)|SL(2,R)]]&nbsp;=&nbsp;{''g''&nbsp;∈&nbsp;M(2,R)&nbsp;:&nbsp;det(''g'')&nbsp;=&nbsp;1}, the [[special linear group]]. Given the profile above, every such  ''g''  lies in a commutative subring P<sub>''m''</sub> representing a type of complex plane according to the square of  ''m''.  Since ''g g''*&nbsp;=&nbsp;''I'', one of the following three alternatives occurs:
* ''mm'' = &minus;''I'' and ''g'' is on a circle of Euclidean [[Rotation (mathematics)|rotations]]; or
* ''mm'' = ''I'' and ''g'' is on an hyperbola of [[squeeze mapping]]s; or
* ''mm'' = 0 and ''g'' is on a line of [[shear mapping]]s.

Writing about [[affine group#Planar affine group|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<sub>''m''</sub> described in the above profile.
The concept of [[identity component]] of the [[group of units]] of P<sub>''m''</sub> leads to the [[polar decomposition]] of elements of the group of units:
*If ''mm'' = &minus;''I'', then ''z'' =&nbsp;ρ&nbsp;exp(θ''m'').
*If ''mm'' = 0, then ''z'' =&nbsp;ρ&nbsp;exp(s&nbsp;''m'')  or ''z'' =&nbsp;&minus; ρ&nbsp;exp(s&nbsp;''m'').
*If ''mm'' = &nbsp;''I'', then ''z'' =&nbsp;ρ&nbsp;exp(''a&nbsp;m'') or ''z'' =&nbsp;&minus;ρ&nbsp;exp(''a&nbsp;m'') or ''z'' =&nbsp;''m''&nbsp;ρ&nbsp;exp(''a&nbsp;m'') or ''z'' =&nbsp;&minus;''m''&nbsp;ρ&nbsp;exp(''a&nbsp;m'').
In the first case exp(θ&nbsp;''m'') =&nbsp;cos(θ)&nbsp;+&nbsp;''m''&nbsp;sin(θ). In the case of the dual numbers exp(''s&nbsp;m'') =&nbsp;1&nbsp;+&nbsp;''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&nbsp;m'') =&nbsp;cosh&nbsp;''a''&nbsp;+&nbsp;''m''&nbsp;sinh&nbsp;''a''.
 
Now <math>\sqrt {\rho \ \exp (a m)} = \sqrt {\rho} \ \exp (a m /2)</math> 
regardless of the subplane P<sub>''m''</sub>, 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&nbsp;exp(''a&nbsp;m'') is an element of the identity component of the group of units of a plane associated with 2&nbsp;&times;&nbsp;2 matrix&nbsp;''m'', then the logarithm function results in a value log&nbsp;ρ&nbsp;+&nbsp;''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<sub>''m''</sub> must be excluded in the cases ''mm'' =&nbsp;0 or ''mm'' =&nbsp;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 ==
<!-- "Dual number" links here -->
Every 2 × 2 real matrix can be interpreted as one of three types of complex number: standard [[complex number]]s, [[dual number]]s, and [[split-complex number]]s. 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]] [[subring]]s ''P''<sub>''m''</sub>. We can determine to which complex plane a given 2&nbsp;×&nbsp;2 matrix belongs as follows and classify which kind of complex number that plane represents.

Consider the 2&nbsp;×&nbsp;2 matrix
:<math> z = \begin{pmatrix}a & b \\ c & d \end{pmatrix}.</math>
We seek the complex plane ''P''<sub>''m''</sub> 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. [[projection (linear algebra)|Projecting]] ''z'' alternately onto these subspaces of R<sup>4</sup> yields
:<math> z = x I + n ,\quad x = \frac{a + d}{2}, \quad n = z - x I .</math>
Furthermore, 
:<math>n^2=pI</math> where <math>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>q = 1/\sqrt{-p}, \quad m = qn</math>. Then <math>m^2= - I, \quad z = x I + m \sqrt{-p}</math>.
*If ''p'' = 0, then it is the [[dual number]]: 
::: <math>z=xI+n</math>. 
*If ''p'' > 0, then ''z'' is a [[split-complex number]]:
:: Let <math>q=1/\sqrt{p}, \quad m = q n</math>. Then <math>m^2 = + I, \quad z = x I + m \sqrt{p}</math>.

Similarly, a 2&nbsp;×&nbsp;2 matrix can also be expressed in [[polar decomposition#Alternative planar decompositions|polar coordinates]] with the caveat that there are two [[connected space|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&ndash;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 .

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[[Category:Affine geometry]]
[[Category:Functions and mappings]]
[[Category:Linear algebra]]
[[Category:Quaternions]]
[[Category:Algebras]]
[[Category:Area]]
[[Category:Matrices]]