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## Matrix convexity

Let be a complex vector space. A set of matrices over consists of subsets for all . Subsets can be considered as sets of matrices over by putting for . More generally, if for some the corresponding set is not given a priori, put this as the empty set. For sets of matrices, we have the following notions of convexity:60

A set of matrices over is called matrix convex or a matrix convex set [Wit84b] if for all and

,

and for all and with

.

A set of matrices over is called absolutely matrix convex [EW97a] if for all and

,

and for all , and with ,

.

A set of matrices over is called a matrix cone [Pow74] if for all and

,

and 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.

#### Footnotes

... convexity:60
With the notation , a set of matrices over is simply a subset of . The sets can be regained as .

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

Subsections

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Prof. Gerd Wittstock 2001-01-07