\(p\)-almost Hadamard matrices and \(\lambda\)-planes (Q2114776)

A square matrix \(H_n\) of degree \(n\) is called \textit{almost Hadamard} if \(U_n:=\frac{H_n}{\sqrt n}\) is orthogonal and it locally maximizes the \(1\)-norm on \(\mathrm{O}(n,\mathbb R)\). A matrix is called \textit{\(p\)-almost Hadamard} for \(p\in[1,2)\) if it locally maximizes the \(p\)-norm on \(\mathrm{O}(n,\mathbb R)\), whereas for \(p\in(2,\infty)\), if it locally minimizes the \(p\)-norm on \(\mathrm{O}(n,\mathbb R)\). The author discusses incidence matrices of \(\lambda\)-planes, i.e., finite projective planes, biplanes, triplanes, and examines which of them are \(p\)-almost Hadamard matrices. He also discusses when orthogonal matrices corresponding to these planes are local minima for the problem: \[\min_{X\in\mathbb R^{n\times n}}\sum_{i,j=1}^n|a_{ij}|^p \text{ where } \sum_{k=1}^na_{ik}a_{kj}=\begin{cases}1\text{ if }i=j\\ 0\text{ otherwise,}\end{cases} \] where \(p>2\).