# Elementary Divisor

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The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

## Elementary Divisor

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), 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.

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
References in periodicals archive ?
be the Smith normal form of matrix A, where U [member of] GL(m, R) and V [member of] GL(n, R).
For the matrix A there exist matrices U [member of] GL(m, R) and V [member of] GL(n, R) such that UAV = [S.sub.A] = diag ([a.sub.1], [a.sub.2], ..., [a.sub.r], 0, ..., 0) is the Smith normal form of A.
be the Smith normal form of matrix A, where S(A) = diag ([a.sub.1], [a.sub.2], ..., [a.sub.r)] [member of] [R.sup.rxr].
Theorem 2.4 (Smith normal form ([6], [section]2.4.4)) Let A [member of] [Z.sub.nxm] be an integer matrix of rank s.
The matrix A' is called the Smith normal form of A.
Proof: Let rank(A) = s and A' = UAV be the Smith normal form of A.

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