Normally related \(n\)-planes and isoclinic \(n\)-planes in \(R^{2n}\) and strongly linearly independent matrices of order \(n\)
Normally related \(n\)-planes in \(R^{2n}\), \(n\geq 2\) and the relationship between sets of pairwise normally related \(n\)-planes and sets of mutually isoclinic \(n\)-planes in \(R^{2n}\) are studied. Two planes \(A\) and \(B\) are normally related in \(R^{2n}\) if \(A\cap B=\{0\}\), \(A\cap B^{\perp}=\{0\}\) and two planes \(C\) and \(D\) are said to be isoclinic if the angle between any nonzero vector in \(C\) and its projection on \(D\) is the same for every nonzero vector in \(C\). It is proved that two normally related \(n\)-planes in \(R^{2n}\) determine an orientation. Some analytic properties for normally related \(n\)-planes are shown. The main result is that: ``The largest maximal isoclinic set or sets in each \(R^{2n}\) are of dimension \(\rho(n)\)'', where \(\rho(n)\) is the Radon-Hurwitz number associated with \(n\). This result is a link between the theorem of \textit{J. F. Adams} [Ann. Math. (2) 75, 603--632 (1962; Zbl 0112.38102)] on the maximal number of strongly linearly independent matrices of order \(n\). The relations between these two results are discussed in detail.
scientific article; zbMATH DE number 4181519