A New Perspective to Algebraic Characterization on Controllability of Multiagent Systems

Algorithm 1

Algorithm for solving invariant factors.

	Step 1: calculate the rank of A(λ). Suppose that rank(A(λ)) = r; then, there are r determinant factors in A(λ).
	Step 2: by elementary transformation for A(λ), choose i1, i2, …, ik rows and i1, i2, …, ik columns (1 ≤ i1 < i2 < ⋯ < ik ≤ r) from A(λ) to constitute a determinant of order k, that is, .
	Step 3: when i1 = 1, i2 = 2, …, ik = k, d1(λ), d2(λ), …, dk(λ) is the greatest common factor of all the determinant of k-order in Step 2.
	Step 4: by step 3, A(λ) turns into its Smith standard form Λ, and it is easy to get the k-order determinant factor of A(λ) as Dk(λ) = d1(λ)d2(λ)…dk(λ), d(k = 1, 2, …, r).
	Step 5: by step 4, let d1(λ) = D1(λ), d2(λ) = D2(λ) ∣ D1(λ), …, dr(λ) = Dr(λ) ∣ Dr−1(λ).