The algebra of hyperboloids of revolution

If we change the sign of p ? m columns (or rows) of an m × m positive definite symmetric matrix A, the resultant matrix B has p negative eigenvalues. We give systems of inequalities for the eigenvalues of B and of the matrix obtained from B by deleting one row and column. To obtain these, we first develop characterizations of the eigenvalues of B which are analogous to the minimum-maximum properties of the eigenvalues of a symmetric A, i.e. the Courant-Fischer theorem. These results arose from studying probability distributions on the hyperboloid of revolution

$\text{x}{}^2{}_1\text{+ ? + x}{}^2{}_{\mathrm{m?p}}\text{? x}{}^2{}_{\mathrm{m\; ?\; p\; +\; 1}}\text{? ? ? x}{}^2{}_{\mathrm{m}}\text{= 1}$

. By contrast, the familiar results are associated with the sphere x²₁ + ? + x²_m = 1.