Knowlpedia

Definition

Conical combination

A finite linear combination with nonnegative coefficients, without a normalization on their sum.

A conical combination of vectors v1,…,vmv_1,\ldots,v_m in a real vector space is a

∑j=1majvj,aj≥0.\sum_{j=1}^m a_jv_j,\qquad a_j\ge0.

The coefficients need not sum to one. The empty combination is zero.

Comparison with convex combinations

If s=∑jaj>0s=\sum_j a_j>0, the combination is ss times the convex combination ∑j(aj/s)vj\sum_j(a_j/s)v_j. Nonnegative coefficients may include zeros; strictly positive coefficients specify a smaller set of representations when generators are fixed.

References