An Argand Diagram for Two by Two Matrices

7/28/2019 An Argand Diagram for Two by Two Matrices

1/9

An Argand Diagram for Two by Two Matrices

Author(s): Tony CrillyReviewed work(s):Source: The Mathematical Gazette, Vol. 87, No. 509 (Jul., 2003), pp. 209-216Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/3621036 .

Accessed: 28/07/2012 20:29

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms

of scholarship. For more information about JSTOR, please contact [email protected].

.

The Mathematical Association is collaborating with JSTOR to digitize, preserve and extend access to The

Mathematical Gazette.

http://www.jstor.org
http://www.jstor.org/action/showPublisher?publisherCode=mathashttp://www.jstor.org/stable/3621036?origin=JSTOR-pdfhttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/stable/3621036?origin=JSTOR-pdfhttp://www.jstor.org/action/showPublisher?publisherCode=mathas


2/9

AN ARGANDDIAGRAM ORTWOBYTWOMATRICES 209An Arganddiagramfor two by two matrices

TONY CRILLYA vivid memoryof mine is of being shown thatcertain2 x 2 matriceswere 'really' complex numbers. The correspondencebetween matrices ofthe form

w = (a -bb aandcomplexnumbers

w = a + ibseemedmagicaland the additionalremark hat

detW = a2 + b2 = Iw12was an unexpected bonus. The correspondencehas been examined invarious articlesin the Gazette[1, 2]. That it is an isomorphism s seen bycomparing

a -b c -d a + c -(b + d)b a d c ,b + d a + c

with(a + ib) + (c + id) = (a + c) + i(b + d),

in the case of addition,and,in the case of multiplication,by comparinga -b 1c -d ac - bd -(ad + bc)ba ad+ b ac - bd

with (a + ib)(c + id) = (ac - bd) + i(ad + bc).For a numericalexercise, we note W1 = 2 ) correspondsto the

complex numberwl = 1 + 2i, andW2 = 1 to 2 = 3 - i. It canbe readily checked that WI + W2corresponds o wl + w2 and that W1W2corresponds o wlw2.

This isomorphismallows us to representa 2 x 2 matrix of the type W(which we call a single complexnumbermatrix in what follows) as a pointon the Argand diagram (Figure 1). For instance Wl = 2correspondingo wl = 1 + 2i maybe representedas a pointin the planewith


3/9

210 THEMATHEMATICALAZErlITEx-y coordinates(1, 2). A matrix of the form 0 (a scalar matrix) isrepresentedby a point (a, 0) on the x-axis and in general any singlecomplex numbermatrix W = ( -b is representedby the point withx-yb acoordinates a, b).

Ywl = 1 + 2i

x

FIGURE:Thetraditionalrgand iagramordisplayingomplexnumbersIn this article we show how this 'matrix - complex number'

isomorphismcan be extended to all 2 x 2 matrices(with real entries)andhow this can be used to framea pictorialrepresentation f these matricesonan 'Arganddiagram'.Thegeneral 2 x 2 matrixAs a first step, we observe thatany 2 x 2 matrixcan be writtenas thesumof two matrices:

a y la -b y xA= a + .i /~6~ b a x -y

Here a = a + y and 6 = a - y requiringthat a = j(a + 6) andy = (a - 6), and similarly b = ?(3- y), x = ?(/ + y). Hence thedecompositionof A is uniquethoughthe second matrix is not a truesinglecomplexnumbermatrix ike the first. However,by introducingE i0 1

we havea y a -b+Ex -

; , ( b a y x 'so thatA = W + EZ

wherebothW andZ aresingle complexnumbermatrices.


4/9

ANARGANDDIAGRAMORTWOBY TWOMATRICESThere is thus an isomorphic correspondenceA


5/9

THEMATHEMATICALAZEIlTEThus the matrixAo corresponds o (4 - 5i) + e (6 + 2i). We note that ATcorresponds o (4 + 5i) + e (6 + 2i) and Aol = is representedby (4 + 5i) + e(-6 - 2i).WilliamKingdon CliffordEntitiesof the form w + ez were studiedby the EnglishmathematicianW. K. Clifford (1845-1879) around the time in the 1870s when he wasreading and translatingHermannGrassmann'sAusdehnungslehre. Thisworkdeals withhigherdimensionalgeometrythrough inearalgebra,a workwhich was largelyneglectedwhen it was first published n 1841 and when asecond edition appeared in 1861. Clifford was thinking of w, z asquaterions and hence his designation 'biquaterions' for forms w + ez (oras he also expressedthem,as motors which were the sum of two rotors) [3,p. 196; 4, pp. 188-189]. As a departure rom the way we are consideringthem, Clifford assumed that the symbol e would commutewith both w andz. The correspondencebetween 2 x 2 matricesand linearalgebras s a richmathematical eam:expressing2 x 2 matrices n a differentway to the onewe haveused leadsto split quaternions 5].TheextendedArgand diagramFor this extension we representa matrix A by its pair of complexnumbers (w, z), and these are plotted individually in the plane as usual(Figure2). Since we aredealingwith orderedpairsof complexnumberswemust distinguishw from z in the diagram(we use a continuousline for wanda dashed ine to indicatethepositionof z).

YA z/

\ \ / x

FIGURE:Argand iagramor2 x 2 matricesWe note thata matrixA is singular f, andonly if, its correspondingpair(w, z) lie on the circumferenceof the same circle since in this case, withk = detA, we have

k = Iw1 - z12 = 0.

212


6/9

AN ARGAND DIAGRAMFOR TWOBY TWO MATRICESCommuting x 2 matricesThe complex numberrepresentation f 2 x 2 matrices as orderedpairsof complex numbersoffers a way of considering commutingmatrices. ForA, Ao representedby w + ez, wo + ezorespectively,we obtainconditionsfor A to commute withAo.

Through the 'matrix


7/9

THEMATHEMATICALAZEI'Eobtain z = z 2 Zoand hence z = mzo where m = 2 is well definedanda real number.

Multiplying he secondrequirement y z3we obtainImw = 1-2 Imw = 12Imwo,zo 1'm wooI2

thatis, Im w = m ImWo.Case (ii) is of specialinterestandyields:

Theorem2. If Aois not a single complexnumbermatrix, henAAo = AoAifandonly if z = mzo(z is 'in line' with zo),wherem is a real numbergivenby Imw = m mwo.The above discussion allows us to pinpointmatricesrepresentedon the'Argand diagram' (Figure 3) which commute with a given matrix Ao.Moreover, t is straightforwardo constructsuch matrices.

A Az/// zo/

, x

W W W W WFIGURE:Argand iagramhowingheconstructionf commuting atricesWe are to construct A whose correspondentw + ez commutes withwo+ ezo, a procedurewe illustrate with a numericalexample. Recall the

matrix Ao = 61 ) For this Ao we have already calculated thatwo = 4 - 5i and zo = 6 + 2i. We now find w + ez wherew = a + biandz = x + yi. Since Imw = m Imwo, we have b = -5m, andbecausez = mzo,z = m(6 + 2i). On the Arganddiagram Figure3), the familyofcomplex numbersw = a - 5mi lies along a horizontalstraight ine (for agiven choice of m) while z lies on the line throughthe origin and passingthrough zo. In this example, w + ez = (a - 5mi) + e(6m + 2mi).Convertingw + ez to matrix orm,we obtain

A = (-a Sm + E 6m -2m = a + 2m llmA -5m a m a - 2m5m a 2m 6m m a - 2m'

214


8/9

AN ARGANDDIAGRAM ORTWOBY TWOMATRICESthe family of matrices which commutes with the given Ao. For example,'7 22A = is one matrix which commutes with Ao and the matrix2 -1

5 33A = is another.,3 -7Conclusion

It is possible to representany 2 x 2 matrixwith real entriesby w + ezin the planein a way reminiscentof the ordinaryArganddiagram. Thusanysuch matrixis in effect a double complex numbermatrixin distinctiontomatrices of the form W = b) which are single complex numberb amatrices. Care must be taken in representingarbitrarymatrices since theorder of the complex numbersw, z is importantand w + ez and z + ewrepresent different matrices in general. If preferred,a matrix can berepresentedon a pair of orthodoxArganddiagramssimultaneously. It maybe interestingto investigatethis furtherwith one diagram representing hecomplex number w and the other z, noting that representinga singlecomplexnumbermatrixW reduces to a single Arganddiagram since in thiscasez = 0).Further nvestigations

Using two Argand diagramsside by side may be helpful in plotting'Mandelbrotype' sets generatedby the matrix terationA - A2 + C,

an immediategeneralisationof the ordinaryMandelbrot terationz - z2 + c.Anotherexploitationof the extendedArgand diagramcould be madeinthe case of that other significantmathematicaldiscovery of the nineteenthcentury, he algebraof quaternions.These areexpressionsof the form

q = a + bi + cj + dkandtheytoo canbe describedas orderedpairsof complexnumbers

q = (a + bi) + (c + di)j.The symbol j plays the same role as e and whereas zj = jz*, like e, thedifferencewithquaternionss that 2 = -1.We have considered2 x 2 matricesfrom an elementaryviewpointbutthere is also an isomorphismbetween the algebraof 2 x 2 matriceswithreal number entries and a sub-algebraof 2 x 2 matrices with complexnumberentries:

a y w z,


9/9

THEMATHEMATICALAZETTEThis lastrepresentation ives aneasy way to handle the multiplicationof theexpressionsw + ez. In the case of quaterions, the isomorphism s

w -Z*q - w + z (w- Z W*andthe twin ArgandDiagrams n this case may be utilised for investigatingthe Mandelbrot quaterion mapping q --- q2 + c.AcknowledgementI would like to thankColin Fletcher,Lans Aleeson and the refereeforhelpfulcomments made on an earlierdraftof this article.References1. FrankGerrish,OrderedPairs,Math.Gaz.79 (March1995)pp. 30-46.2. Douglas Quadling, Q is for Quaterions, Math. Gaz. 63 (June 1979)pp. 98-110.3. William K. Clifford, A preliminary sketch of biquaternions,Proc.London Math.Soc., 4 (1873) pp. 381-395. Also Coll. Papers, (Reprint:1968) pp. 181-200.4. B. VanderWaerden,A history of algebra,Springer 1985).5. B. A. Rosenfel'd,tr. A. Shenitzer,A history of non-Euclideangeometry,Springer 1988). TONY CRILLY

MiddlesexBusinessSchool, TheBurroughs,Hendon,LondonNW4 4BTe-mail:[email protected] irrational

Sines,cosinesandtangentsweremy constantompanionsn my careeras anengineer. oachieve quare ndcircular ake insof equalarea equireshat hesidelengthof thesquareinmustbeexactly88.64% f the diameter f theround ne.Anumerateersonwould ememberhisratiobyrote.

Nick Lordsaw this in a letterto the Daily Telegraphn April2003. Theimplications that = 3.14281984 recisely.

Exams remorerecent han outhinkThe CambridgeMathematicalriposof 1904(possibly he oldestuniversityexaminationntheworld)...This unintendedmplicationromthe Gazetteof March2003 madeBryanThwaiteslad oknow hathe satthefortiethet of papers

216

An Argand Diagram for Two by Two Matrices

Documents

Transcript of An Argand Diagram for Two by Two Matrices