Elementary Divisor

The elementary divisors of a square matrix are the polynomials (λ – λ₁)^p1, (λ – λ₂)^p2, . . ., (λ – λ_s)^p_s obtained from the characteristic equation

The minors of the kth-order determinant Δ(λ), k ≤ n, are polynomials in λ. Let D_k (λ), k = 1, 2,. . ., n, be the greatest common divisor of these polynomials, with D_n (λ) ≡ Δ(λ). In the sequence

D₀(λ) ≡ 1, D₁(λ), D₂(λ),...,D_n(λ)

each polynomial is divisible by the preceding one without remainder. Let the corresponding quotients be expressed as a product of linear factors in the field of complex numbers:

The polynomials (λ – λ′)^a1, (λ – λ″)^a2, . . ., (λ – λ′)^l1, (λ – λ″)^l2, . . . form the complete system of elementary divisors of A (here, powers with zero exponents are not taken into consideration).

The product of all elementary divisors is equal to the characteristic polynomial. The elementary divisors determine the Jordan form of A.