In mathematics, a
P
-matrix is a complex square matrix with every principal minor > 0. A closely related class is that of
P
0
-matrices, which are the closure of the class of
P
-matrices, with every principal minor
≥
0.
By a theorem of Kellogg, the eigenvalues of
P
- and
P
0
- matrices are bounded away from a wedge about the negative real axis as follows:
If
{
u
1
,
.
.
.
,
u
n
}
are the eigenvalues of an
n
-dimensional
P
-matrix, where
n
>
1
, then
|
a
r
g
(
u
i
)
|
<
π
−
π
n
,
i
=
1
,
.
.
.
,
n
If
{
u
1
,
.
.
.
,
u
n
}
,
u
i
≠
0
,
i
=
1
,
.
.
.
,
n
are the eigenvalues of an
n
-dimensional
P
0
-matrix, then
|
a
r
g
(
u
i
)
|
≤
π
−
π
n
,
i
=
1
,
.
.
.
,
n
The class of nonsingular M-matrices is a subset of the class of
P
-matrices. More precisely, all matrices that are both
P
-matrices and Z-matrices are nonsingular
M
-matrices. The class of sufficient matrices is another generalization of
P
-matrices.
The linear complementarity problem
L
C
P
(
M
,
q
)
has a unique solution for every vector
q
if and only if
M
is a
P
-matrix.
If the Jacobian of a function is a
P
-matrix, then the function is injective on any rectangular region of
R
n
.
A related class of interest, particularly with reference to stability, is that of
P
(
−
)
-matrices, sometimes also referred to as
N
−
P
-matrices. A matrix
A
is a
P
(
−
)
-matrix if and only if
(
−
A
)
is a
P
-matrix (similarly for
P
0
-matrices). Since
σ
(
A
)
=
−
σ
(
−
A
)
, the eigenvalues of these matrices are bounded away from the positive real axis.
