Cone (linear algebra)

In linear algebra, a (linear) cone is a subset of a vector space that is closed under multiplication by positive scalars. In other words, a subset C of a real vector space V is a cone if and only if λx belongs to C for any x in C and any positive scalar λ of V (or, more succinctly, if and only if λC = C for any positive scalar λ).

A cone is said to be pointed if it includes the null vector (origin) 0; otherwise it is said to be blunt. Some authors use "non-negative" instead of "positive" in this definition of "cone", which restricts the term to the pointed cones only.

The definition makes sense for any vector space V which allows the notion of "positive scalar" (i.e., where the ground field is an ordered field), such as spaces over the rational, real algebraic, or (most commonly) real numbers.

The concept can also be extended for any vector space V whose scalar field is a superset of those fields (such as the complex numbers, quaternions, etc.), to the extent that such a space can be viewed as a real vector space of higher dimension.

Related concepts

The cone of a set

The (linear) cone of an arbitrary subset X of V is the set X^* of all vectors λx where x belongs to X and λ is a positive scalar.

With this definition, the cone of X is pointed or blunt depending on whether X contains the origin 0 or not. If "positive" is replaced by "non-negative" in this definition, then the cone of X will be pointed, for any X.

Salient cone

A cone X is said to be salient if it does not contain any pair of opposite nonzero vectors; that is, if and only if C$\cap$(-C) $\subseteq$ {0}.

Convex cone

A convex cone is a cone that is closed under convex combinations, i.e. if and only if αx + βy belongs to C for any non-negative scalars α, β with α + β = 1.

Affine cone

If C - v is a cone for some v in V, then C is said to be an (affine) cone with vertex v. More commonly, in algebraic geometry, the term affine cone over a projective variety X in PV is the affine variety in V given as the preimage of X under the quotient map

$V\setminus\{0\}\to \mathbf{P}V.$

Proper cone

The term proper cone is variously defined, depending on the context. It often means a salient and convex cone, or a cone that is contained in an open halfspace of V.

Properties

Boolean, additive and linear closure

Linear cones are closed under Boolean operations (set intersection, union, and complement). They are also closed under addition (if C and D are cones, so is C + D) and arbitrary linear maps. In particular, if C is a cone, so is its opposite cone -C.

Spherical section and projection

Let |·| be any norm for V, with the property that the norm of any vector is a scalar of V. Let S be the unit-norm sphere of V, that is, the set

$S = \{\, x \in V\;:\; |x| = 1 \,\}$

By definition, a nonzero vector x belongs to a cone C of V if and only if the unit-norm vector x/|x| belongs to C. Therefore, a blunt (or pointed) cone C is completely specified by its central projection onto S; that is, by the set

$C' = \bigg\{\, \frac{x}{|x|} \;:\; x \in C \wedge x \neq \mathbf{0} \,\bigg\}$

It follows that there is a one-to-one correspondence between blunt (or pointed) cones and subsets of S. Indeed, the central projection C' is simply the spherical section of C, the set C$\cap$S of its unit-norm elements.

A cone C is closed with respect to the norm |·| if it is a closed set in the topology induced by that norm. That is the case if and only if C is pointed and its spherical section is a closed subset of S.

Note that the cone C is salient if and only if its spherical section does not contain two opposite vectors; that is, C' $\cap$(-C' ) = {}.

See also

• Cone (disambiguation)
  • Cone (geometry)
  • Cone (topology)
  • Convex cone
• Ordered group with the concept of the "positive cone"
• Ordered vector space

References