Elementary Divisor

Elementary Divisor

 

The elementary divisors of a square matrix Elementary Divisor 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), kn, 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.

References in periodicals archive ?
There is a compact formula for the elementary divisors of the Cartan matrix [C.
Yamada, Elementary divisors of Cartan matrices for symmetric groups J.