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Statement

Definition (pp. 88--89). An $n$-rowed square matrix $A$ with every entry $0$ or $1$ is an incidence matrix when (I1) any two distinct rows both have a $1$ in exactly one common column, (I2) any two distinct columns both have a $1$ in exactly one common row, and (I3) every row has at least three ones.

Theorem 2 (p. 89, quoted). "If $\pi$ is a finite projective plane geometry with $N+1$ points on a line, then there exists an incidence matrix $A$ of order $n=N^2+N+1$. If $A^{\mathrm T}$ denotes the transpose of the matrix $A$, then

$$\text{(M)}\qquad B=AA^{\mathrm T}=A^{\mathrm T}A,$$

where $B$ is an integral matrix with $N+1$ down the main diagonal and ones in all other positions."

In later notation $B=NI+J$, with $I$ the identity and $J$ the all-ones matrix of order $n$; the paper does not write it that way.

Proof pointer

P. 89. Number the points and the lines of the plane $1,\ldots,N^2+N+1$ and put a $1$ in row $i$, column $j$ exactly when line $i$ contains point $j$. The plane's axioms give (I1)--(I3) and the equation (M).

Read depth. Claims checked: the definition and the theorem were read clause by clause on the page image of the print, and the proof was followed. Nothing here is independently reviewed.

Dependencies

None in the corpus. It is the first step of the proof of Theorem 1; Theorem 3 is its converse.

Source. R. H. Bruck and H. J. Ryser, The nonexistence of certain finite projective planes, Canad. J. Math. 1 (1949), 88--93, doi:10.4153/CJM-1949-009-2; the edition read is named on the source card.

Bears on

• Problem 723: the problem asks whether every finite projective plane has prime-power order. Theorem 2 turns a plane of order $N$ into a $0$--$1$ matrix solution of (M), the object Theorem 1's arithmetic argument rules out for the excluded orders; on its own it excludes no order.