Pattern Recognition

Equation (2.170) reveals an important observation. The eigenvalue zero (λ= 0) can occur for two reasons: either the right side is a non-trivial linear combination, i.e., at least two coefficients (∗) are nonzero, or all coefficients equal zero. In the latter case, the eigenvector v is perpendicular to every ϕ(mk). But if the eigenvalue is nonzero (λ ≠ 0), then the right side must be a nonzero linear combination. Hence, v is a non-trivial linear combination of the ϕ(mk). To summarize, any eigenvector solution of Equation (2.169) that corresponds to a nonzero eigenvalue is in the span of {ϕ(m1), . . . , ϕ(mN)}.

Hence, in case λ ≠ 0, one can rewrite v as a linear combination of the ϕ(m1), . . . , ϕ(mN) (see Equation (2.171)) or as the product ...