A set of matrices over
is called
matrix convex or a matrix convex
set
[Wit84b] if for all
and
A set of matrices over
is called
absolutely matrix convex [EW97a] if for all
and
A set of matrices over
is called a
matrix cone [Pow74] if for all
and
A set of matrices over
is matrix convex if and only if all
matrix convex combinations of elements of
are again in
.
is absolutely
matrix convex if and only if all
absolutely matrix convex combinations of elements of
are again in
.
Here, a
matrix convex combination of
, ...,
(
) is a sum of the form
with matrices
such that
. An
absolutely matrix convex combination of
, ...,
is a sum
of the form
with matrices
and
such that
and
.
If is a topological vector space, topological terminology is to be considered at
all matrix levels:
For instance, a set
of matrices over
is called
closed if all
are closed.
Then the definitions have the following form:
Let
.
is called matrix convex if
is called absolutely matrix convex if
is called a matrix cone if