The elementary divisors of a square matrix are the polynomials (λ – λ1)p1, (λ – λ2)p2, . . ., (λ – λs)ps obtained from the characteristic equation
The minors of the kth-order determinant Δ(λ), k ≤ n, are polynomials in λ. Let Dk (λ), k = 1, 2,. . ., n, be the greatest common divisor of these polynomials, with Dn (λ) ≡ Δ(λ). In the sequence
D0(λ) ≡ 1, D1(λ), D2(λ),...,Dn(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.