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