If all elementary divisors are linear, " A " is diagonalizable.
2.
The elementary divisors of a square matrix " A " are the characteristic polynomials of its Jordan blocks.
3.
The factors of the minimal polynomial " m " are the elementary divisors of the largest degree corresponding to distinct eigenvalues.
4.
The degree of an elementary divisor is the size of the corresponding Jordan block, therefore the dimension of the corresponding invariant subspace.
5.
Conversely, knowing the multiset of elementary divisors, the invariant factors can be found, starting from the final one ( which is a multiple of all others ), as follows.
6.
The module is determined up to isomorphism by specifying its free rank, and for class of associated irreducible elements and each positive integer the number of times that occurs among the elementary divisors.
7.
Note that in a PID, the nonzero primary ideals are powers of prime ideals, so the elementary divisors can be written as powers q _ i = p _ i ^ { r _ i } of irreducible elements.
8.
The elementary divisors can be obtained from the list of invariant factors of the module by decomposing each of them as far as possible into pairwise relatively prime ( non-unit ) factors, which will be powers of irreducible elements.
9.
"' Ludwig Stickelberger "'( May 18, 1850 & ndash; April 11, 1936 ) was a Swiss mathematician who made important contributions to linear algebra ( theory of elementary divisors ) and algebraic number theory ( Stickelberger relation in the theory of cyclotomic fields ).
10.
The list of primary ideals is unique up to order ( but a given ideal may be present more than once, so the list represents a multiset of primary ideals ); the elements q _ i are unique only up to associatedness, and are called the " elementary divisors ".
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