# Elementary Divisor

## Elementary Divisor

The elementary divisors of a square matrix are the polynomials (λ – λ_{1})^{p1}, (λ – λ_{2})^{p2}, . . ., (λ – λ_{s})^{ps} obtained from the characteristic equation

The minors of the *k*th-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*(λ) ≡ Δ(λ). In the sequence

_{n}*D*_{0}(λ) ≡ 1, *D*_{1}(λ), *D*_{2}(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.*