Cone (linear algebra)

In linear algebra is a (linear ) cone is a subset of a vector space that is closed under multiplication by positive scalars.

• 3.1 cone shell
• 3.2 Dual Cone
• 3.3 Polar cone

Definition

Be a parent body, such as the real or rational numbers. A subset of a vector space hot (linear ) cone, if for each element and each non -negative scalar too.

An equivalent characterization is: A subset of a vector space is exactly then a (linear ) cone, if for all non-negative scalar.

Types of cones

Acute and obtuse cone

A cone is called pointed if it contains no straight line, that is, otherwise dull.

Dotted cone

Some authors restrict the above definition to the seclusion under multiplication with true positive scalars. In this case, let dotted cone (ie, which is not included ) and plug with 0 differ.

Convex cone

A convex cone is a cone, which is closed with a linear combination of non-negative coefficients. is thus convex cone if and only if.

Convex cones play an important role in the linear optimization.

Affine cone

If for one and a cone, it is called ( affine ) cone with vertex. Clearly that is a (linear ) cone along the position vector is moved.

Operators

Cone-shaped shell

The cone-shaped shell of any subset is defined by

This is apparently the smallest cone containing all.

It should be noted that actually satisfies the definition of an envelope operator.

Dual cone

Specifies the dual space of and is further a cone in. Then the dual cone is defined by

It denotes the dual pairing, that is, it is.

Is even a Hilbert space with scalar product - this is not a priori equivalent to the dual pairing - this is the definition of the dual cone simplifies to

Clearly, these are then include all the vectors with all elements of the cone at an angle of 90 °.

Polar cone

Analogously, the term of the polar cone formulate:

In a Hilbert space, then:

So the set of all vectors which have an angle of at least 90 ° cone with all the elements and is therefore considered

For both versions of the definition results in the correlation in the relevant vector space.

Note: The use of the latter two terms is not uniform in the literature. The dual as the polar cone defined, or each second form of the definition used in incomplete Prähilberträumen in order to interpret the resulting sets as a cone in the original space can - So sometimes - especially in the English -speaking world.

Spherical average

If the vector space normalized by, this is how the central projection of a cone on the unit circle look. This is by

Explained. Your picture is obviously equal to the intersection of the cone with the unit circle.

A cone is completely described by its circular section, as it applies:

Properties

• The intersection of a family of cones is a cone, as is the union.
• The complement of a cone is a cone back.
• For two cones and the sum of each cone.
• The dual and the polar cone is convex, regardless of whether this property already approached the original cone or not.
• Is a topological vector space - with the topological dual space - so the polar and dual cones are always complete.