Elementary Divisor

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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.