Let
H
H
be a positive semi-definite
m
n
mn
-by-
m
n
mn
Hermitian matrix, partitioned into
m
2
{m^2}
n
n
-square blocks
H
i
j
,
i
,
j
=
1
,
…
,
m
{H_{ij}},i,j = 1, \ldots ,m
. We denote this by
H
=
[
H
i
j
]
H = [{H_{ij}}]
. Consider the function
f
:
M
n
→
M
r
f:{M_n} \to {M_r}
given by
f
(
X
)
=
X
k
f(X) = {X^k}
(ordinary matrix product) and denote
H
f
=
[
f
(
H
i
j
)
]
{H_f} = [f({H_{ij}})]
. We shall show that if
H
H
is positive semi-definite then under some restrictions on
H
i
j
,
H
f
{H_{ij}},{H_f}
is also positive semi-definite. This generalizes familar results for Hadamard and ordinary products.