Hyperorthogonal family of vectors and the associated Gram matrix (Q482039)

A \(p\)-tuple \((v_{1},\dots,v_{p})\) of nonzero vectors in \(\mathbb{R}^{n}\)\ is called hyperorthogonal if the angle between any two distinct vectors in the list is at least \(\frac{1}{2}\pi\); equivalently, if the positive semidefinite Gram matrix \(A:=[\left\langle v_{i},v_{j}\right\rangle ]_{i,j}\) of inner products has all entries \(\leq0\) off the main diagonal. A hyperorthogonal \(p\)-tuple \((v_{1},\dots,v_{p})\) is called maximal if it cannot be extended by any vector in the subspace spanned by \(v_{1},\dots,v_{p}\), and is called irreducible if this set of vectors cannot be partitioned into two proper subsets which are orthogonal to one another. The rank of the \(p\)-tuple is the dimension of the subspace spanned by \(v_{1},\dots,v_{p}\) (\(=\mathrm{rank}(A)\)). A pair \((v,\alpha v)\) of linearly dependent vectors is hyperorthogonal if and only it is antipodal (\(\alpha<0\)). The following is proved. Let \((v_{1},\dots,v_{p})\) be a hyperorthogonal \(p\)-tuple of rank \(r\). If \((v_{1},\dots,v_{p})\) is irreducible then \(p=r\) or \(r+1\), and any two of the following properties imply the third: (i) \((v_{1},\dots,v_{p})\) is irreducible; (ii) \((v_{1},\dots,v_{p})\) is maximal and (iii) \(p=r+1\). In general, a hyperorthogonal \(p\)-tuple can be uniquely partitioned into sublists which are irreducible hyperorthogonal and where these sublists are mutually orthogonal. Moreover, the hyperorthogonal sublists are all maximal if and only if the original \(p\)-tuple is maximal. If there are \(m\) sublists then \(p\leq r+m\) with equality if and only if the original \(p\)-tuple is maximal. The case where \(p=2n\) occurs exactly when \((v_{1},\dots,v_{2n})\) can be partitioned into \(n\) mutually orthogonal pairs of antipodal vectors\(.\) The author also interprets these results in terms of the associated Gram matrices.