Number Theory: An Introduction to Mathematics

6 Designs 249

A 2-design is said to besquareor (more commonly, but misleadingly) ‘symmetric’
ifb=v, i.e. if the number of blocks is the same as the number of points. Thus any
projective plane is a square 2-design.
For a square 2-(v,k,λ)design,k=rand the incidence matrixAis itself nonsin-
gular. The first relation (5) is now equivalent toJvA=kJv.Sincek=r,thesumof
the entries in any row ofAis alsokand thusJvAt=kJv. By multiplying the second
relation (5) on the left byA−^1 and on the right byA, we further obtain

AtA=(r−λ)Iv+λJv.

ThusAtis also the incidence matrix of a square 2-(v,k,λ)design, thedualof the
given design.
This partly combinatorial argument may be replaced by a more general matrix one:

Lemma 21Let a,b,k be real numbers and n> 1 an integer. There exists a nonsin-
gular real n×n matrix A such that

AAt=aI+bJ, JA=kJ, (6)

if and only if a> 0 ,a+bn> 0 and k^2 =a+bn. Moreover any such matrix A also
satisfies
AtA=aI+bJ, JAt=kJ. (7)

Proof We show first that ifAis any realn×nmatrix satisfying (6), thena+bn=k^2.
In fact, sinceJ^2 =nJ, the first relation in (6) impliesJAAtJ=(a+bn)nJ, whereas
the second impliesJAAtJ=k^2 nJ.
We show next that the symmetric matrixG:=aI+bJis positive definite if and
only ifa>0anda+bn>0. By Lemma 15, detG=an−^1 (a+bn).IfGis positive
definite, its determinant is positive. Since all principal submatrices are also positive
definite, we must haveai−^1 (a+bi)>0for1≤i≤n. In particular,a+b>0,
a(a+ 2 b)>0, which is only possible ifa>0. It now follows that alsoa+bn>0.
Conversely, supposea>0anda+bn>0. Then detG>0 and there exist nonzero
real numbersh,ksuch thata=h^2 ,a+bn=k^2. If we putC=hI+(k−h)n−^1 J,
thenJC=kJand

C^2 =h^2 I+{ 2 h(k−h)+(k−h)^2 }n−^1 J=aI+bJ=G.

Since detG>0, this shows thatG=CCtis positive definite andCis nonsingular.
Finally, letAbe any nonsingular realn×nmatrix satisfying (6). SinceAis nonsin-
gular,AAtis a positive definite symmetric matrix and hencea>0,a+bn>0. Since
AAt=C^2 andCt=C,wehaveA=CU,whereUis orthogonal. HenceAt=UtC
andC=UAt.FromJC=kJwe obtainkJ=JA=JCU=kJU. ThusJ=JU
andJAt=JU At=JC=kJ. MoreoverUtJU=J,sinceJt=J, and hence

AtA=UtC^2 U=Ut(aI+bJ)U=aI+bJ. 

In Chapter VII we will derive necessary and sufficient conditions for the existence
of a nonsingularrational n×nmatrixAsuch thatAAt=aI+bJ, and thus in particular
obtain some basic restrictions on the parametersv,k,λfor the existence of a square
2-(v,k,λ)design. These were first obtained by Bruck, Ryser and Chowla (1949/50).