The algebra of hyperboloids of revolution (Q1063662)

Let A be a positive definite symmetric matrix of order m, \(J^ p_ m\) the diagonal matrix whose first m-p entries are \(+1\) and whose last p entries are -1. Defining the ''hyperbolic transpose'' by \(x^ h=(J^ p_ mx)^ t\), t the usual or ''spherical transpose'', the equation \(\sum^{m- p}_{1}x^ 2_ i-\sum^{m}_{m-p+1}x^ 2_ i=1\) of a hyperboloid of revolution can be written \(x^ hx=1\), such as \(x^ tx=1\) is the equation of a sphere. The matrix \(B=AJ^ p_ m\) results from A by changing the signs of the last p columns. The authors prove that B has m- p positive and p negative eigenvalues and characterize these eigenvalues by minimax properties of \(y^ hBy\) subject to certain eigenvector restrictions, such as the Courant-Fischer theorem characterizes the eigenvalues of A using the quadratic form \(y^ tAy\). Furthermore, they relate the eigenvalues of \(AJ^ p_ m\) and \(AJ_ m^{p+1}\) and of B and \(B_{(r)}\), \(B_{(r)}\) being obtained from B by deleting the \(r^{th}\) row and \(r^{th}\) column.