Why cones are represented by matrices
I see there are multiple definitions of cones:
1) Cone $K$ is defines as a set of vertices $[x_1, x_2, x_3, ...]$ with $[0]$ as the base (starting point)
2) Cone $K$ is defined as intersection of half-spaces $[langle a_i,x rangle leq 0]$
3) Cone $K$ is defined by a matrix, e.g. $K in X^{n}_{+}$, such that $K$ is semi-definite positive.
The 1) and 2) are easy to interpret and draw, since both define the boundaries in the space, the boundaries+interior is the cone itself. But I dont understand how the definition 3) defines the cone (e.g. the shape in the Euclidean space).
1) How do I interpret the matrix as a cone ?
2) How to check/see whether a vector belongs to the cone ?
3) How to derive a dual-cone from a cone defined by a matrix ?
convex-geometry convex-cone dual-cone
asked Nov 29 at 6:10
Martin G
498
add a comment |
I see there are multiple definitions of cones:
1) Cone $K$ is defines as a set of vertices $[x_1, x_2, x_3, ...]$ with $[0]$ as the base (starting point)
2) Cone $K$ is defined as intersection of half-spaces $[langle a_i,x rangle leq 0]$
3) Cone $K$ is defined by a matrix, e.g. $K in X^{n}_{+}$, such that $K$ is semi-definite positive.
The 1) and 2) are easy to interpret and draw, since both define the boundaries in the space, the boundaries+interior is the cone itself. But I dont understand how the definition 3) defines the cone (e.g. the shape in the Euclidean space).
1) How do I interpret the matrix as a cone ?
2) How to check/see whether a vector belongs to the cone ?
3) How to derive a dual-cone from a cone defined by a matrix ?
convex-geometry convex-cone dual-cone
asked Nov 29 at 6:10
Martin G
498
add a comment |
I see there are multiple definitions of cones:
1) Cone $K$ is defines as a set of vertices $[x_1, x_2, x_3, ...]$ with $[0]$ as the base (starting point)
2) Cone $K$ is defined as intersection of half-spaces $[langle a_i,x rangle leq 0]$
3) Cone $K$ is defined by a matrix, e.g. $K in X^{n}_{+}$, such that $K$ is semi-definite positive.
The 1) and 2) are easy to interpret and draw, since both define the boundaries in the space, the boundaries+interior is the cone itself. But I dont understand how the definition 3) defines the cone (e.g. the shape in the Euclidean space).
1) How do I interpret the matrix as a cone ?
2) How to check/see whether a vector belongs to the cone ?
3) How to derive a dual-cone from a cone defined by a matrix ?
convex-geometry convex-cone dual-cone
asked Nov 29 at 6:10
Martin G
498
I see there are multiple definitions of cones:
1) Cone $K$ is defines as a set of vertices $[x_1, x_2, x_3, ...]$ with $[0]$ as the base (starting point)
2) Cone $K$ is defined as intersection of half-spaces $[langle a_i,x rangle leq 0]$
3) Cone $K$ is defined by a matrix, e.g. $K in X^{n}_{+}$, such that $K$ is semi-definite positive.
The 1) and 2) are easy to interpret and draw, since both define the boundaries in the space, the boundaries+interior is the cone itself. But I dont understand how the definition 3) defines the cone (e.g. the shape in the Euclidean space).
1) How do I interpret the matrix as a cone ?
2) How to check/see whether a vector belongs to the cone ?
3) How to derive a dual-cone from a cone defined by a matrix ?
convex-geometry convex-cone dual-cone
convex-geometry convex-cone dual-cone
asked Nov 29 at 6:10
Martin G
498
asked Nov 29 at 6:10
Martin G
498
asked Nov 29 at 6:10
Martin G
498
asked Nov 29 at 6:10
Martin G
498
asked Nov 29 at 6:10
Martin G
498
498
add a comment |
add a comment |
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