Overemphasis on the topic of rational canonical form in section 5 makes the second half of the chapter harder than it needs to be. Instead of rational canonical form, it is better to focus on the fact that, by virtue of Proposition 1 on page 97, every matrix of the form xI - A is equivalent to a diagonal matrix in which each diagonal entry is either 1 or a power of an irreducible monic polynomial. (A monic polynomial can be written as a product of irreducible factors. If two of these factors are distinct, it can be written as a product of two relatively prime factors, in which case Proposition 1 shows the diagonal entry can be split into two diagonal entries of lower degree.)
If A is a square matrix with rational number entries, the elementary divisors of A are the diagonal entries other than 1 of a diagonal matrix equivalent to xI - A whose diagonal entries are powers of irreducible monic polynomials. For this definition to be valid, it must be shown that A determines these diagonal entries, that is, that two diagonal matrices of the required type (diagonal entries either 1 or powers of irreducible monic polynomials) are equivalent if and only if one is obtained from the other merely by rearranging the diagonal entries. This follows from the observation that Proposition 1 implies that the equivalent strongly diagonal matrix is obtained by a simple process of putting together diagonal entries that are powers of different irreducible polynomials and arranging them in an order so that the exponents increase as you go down the diagonal.
When the elementary divisors of A are so defined, the theorems of Sections 2 and 3 immediately imply that A and B are similar if and only if they have the same elementary divisors.
The characteristic polynomial of A (the determinant of xI - A) is clearly the product of the elementary divisors of A. In other words, the elementary divisors of A give a factorization of the characteristic polynomial of A. That is the (rather poor) reason they are called `divisors.'
The theorem of Section 4 constructs a matrix with given elementary divisors.
If D is the strongly diagonal matrix equivalent to xI - A, the elementary divisors are the diagonal entries of the matrix obtained by using Proposition 1 to split the diagonal entries of D into polynomials that are powers of irreducible polynomials. Therefore, the elementary divisors are found by writing the diagonal entries of D as products of powers of irreducibles. From this it follows that the last diagonal entry of D is the least common multiple of the elementary divisors of A. Thus, by the theorem of Section 6, the minimum polynomial of A is the least common multiple of the elementary divisors of A.
The theorem of Section 7 says that A is diagonalizable if and only if its elementary divisors are all of the first degree.
Return to Chapter 9.