On principal sections of a pair of forms

Abstract:

Let H and C be $n \times n$ Hermitian matrices with C positive definite. Let $H({i_1}, \ldots ,{i_r})$ denote the submatrix of H formed by deleting the rows and columns ${i_1}, \ldots ,{i_r}$, of H. In this paper, with ${r_1} + \cdots + {r_k} \leq n$, we study the roots of the determinantal equation $\det (\lambda C - H) = 0$ and those of \[ \det ((\lambda C - H)({r_1} + \cdots + {r_{i - 1}} + 1, \ldots ,{r_1} + \cdots + {r_i})) = 0\] for $i = 1, \ldots ,k$.