Mutual incidence matrix of two balanced incomplete block designs (Q6615595)

A balanced incomplete block design with parameters \((v,b,r,k,\lambda)\) built on a finite set \(\left|V\right|=v\) is a list of \(b\) blocks, each of which is a \(k\)-element subset of \(V\), such that every element of \(V\) is contained in exactly \(r\) blocks and every pair of elements of \(V\) is contained in exactly \(\lambda\) blocks. The incidence matrix of a BIBD is a matrix whose rows correspond to points and columns correspond to blocks. The entry in the row \(i\) and column \(j\) is 1 if the point \(i\) belongs to the block \(j\) and 0 otherwise. The aim of the paper is to study a mutual incidence matrix \(M(D_1, D_2)\) for two block designs \(D_1\), \(D_2\) with parameters \((v,b_i,r_i,k_i,\lambda_i)\) built on the same set \(V\). The rows of the matrix are indexed by blocks of \(D_1\) and the columns are indexed by blocks of \(D_2\). The entry on the intersection of a row and a column is the number of points shared by the corresponding blocks of the two block designs.\N\NThe authors find all eigenvalues of the matrices \(MM^T\) and \(M^TM\) and their eigenspaces in terms of the parameters of the two block designs. By the Theorem 1 the diagonal elements of the matrix \(MM^T\) are equal and their value is:\N\[\N(MM^T)_{nn} = k_1(\lambda_2k_1 - \lambda_2 + r_2), n=1, \ldots, b_1.\N\]\NNext the authors prove that the matrices \(MM^T\) and \(M^TM\) are of rank \(v\) and have only two nonzero eigenvalues \(\mu_1\) and \(\mu_2\):\N\begin{itemize}\N\item \(\mu_1 = r_1r_2k_1k_2\) with the eigenspace spanned by the vector \((1, \ldots, 1)^T\),\N\item \(\mu_2 = (r_1 - \lambda_1)(r_2 - \lambda_2)\) with eigenspace of dimension \(v - 1\) defined by \(V_{D_i}:=\mathrm{Span}\{Z_{D_i}(x,y)|x,y \in V\}\), \(V_{D_1}\) corresponds to the \(MM^T\) and \(V_{D_2}\) to \(M^TM\) respectively.\N\end{itemize}\N\N\(Z_{D_i}(x,y)\) is a vector corresponding to a block design \(D_i\), \(i=1,2\) and is defined in the following way:\N\[\NZ_{D_i}(x,y):=(Z^1_{D_i}(x,y), \ldots, Z^{b_i}_{D_i}(x,y))^T,\N\]\Nwhere\N\[\NZ^i_{D_i}(x,y):= \begin{cases} 0 & \mbox{if } x,y \in {B^j}_i \mbox{ or if } x,y \notin {B^j}_i,\\\N1 & \mbox{if } x \in {B^j}_i \mbox{ and } y \notin {B^j}_i,\\\N-1 & \mbox{if } x \notin {B^j}_i \mbox{ and } y \in {B^j}_i.\\\N\end{cases} \N\]\NThe characteristic polynomial of \(MM^T\) has the form:\N\[\N\det(MM^T-tI)=(-1)^{b_1}t^{b_1-v}(t-\mu_1)(t-\mu_2)^{v-1},\N\]\Nwhile the characteristic polynomial of \(M^TM\) is:\N\[\N\det(M^TM-tI)=(-1)^{b_2}t^{b_2-v}(t-\mu_1)(t-\mu_2)^{v-1}.\N\]