Rational, Block and Diagonal Divisibility Canonical Forms

If A is an r × r matrix and B is an s × s matrix, then their direct sum A ⊕B is the (r + s) × (r + s) matrix A ⊕B = A B .

A rational7 canonical8 form is a matrix R that is a direct sum of companion matrices, R = C(g1) ⊕· · · ⊕C(gt), where the gi(x) are monic polynomials with g1(x) | g2(x) | · · · | gt(x). If a matrix A is similar to the rational canonical form C(g1) ⊕· · · ⊕C(gt), where g1(x) | g2(x) | · · · | gt(x), then we say that the invariant factors of A are g1(x), g2(x), . . . , gt(x). We have just proved that every n ×n matrix over a field is similar to a rational canonical form, and so it has invariant factors. Can a matrix A have more than one list of invariant factors?

A 1 × 1 canonical blocks is a matrix J(α, 1) = [α]. If s ≥2, then an s × s canonical blocks is a matrix J(α, s) of the form J(α, s) =   α · · · α · · · α · · · ... ... ... ... ... ... · · · α · · · α   . . Here is a more compact description of a canonical blocks. Let L denote the s × s matrix having all entries 0 except for 1’s on the subdiagonal just below the main diagonal. In this notation, a canonical blocks J(α, s) has the form J(α, s) = αI + L. Let us regard L as a linear transformation on ks. If e1, . . . , es is the standard basis, then Lei = ei+1 if i < s while Les = 0. It follows easily that the matrix L2 is all 0’s except for 1’s on the second subdiagonal below the main diagonal; L3 is all 0’s except for 1’s on the third subdiagonal; Ls−1 has 1 in the s, 1 position, with 0’s everywhere else, and Ls = 0.

If A is an n × n complex matrix, then eA = ∞ k=0 k! Ak = I + A + 1 2 A2 + 1 6 A3 + · · · . It can be proved that this series converges for every matrix A, and that the function A ↦eA is continuous; that is, if limk→∞Ak = M, then lim k→∞eAk = eM.

Two n × t matrices ! and !′ with entries in a commutative ring R are R-equivalent if there are invertible matrices P and Q with entries in R with !′ = Q!P (writing P is just as general as writing P−1). Of course, equivalence as just defined is an equivalence relation on the set of all (rectangular) n × t matrices over R.

There are three elementary row operations on a matrix A with entries in a commutative ring R: Type I: Multiply ROW( j) by a unit u ∈R. Type II: Replace ROW(i) by ROW(i) + c j ROW( j), where j ̸= i and c j ∈R. Type III: Interchange ROW(i) and ROW( j). There are analogous elementary column operations. Notice that an operation of type III (i.e., an interchange) can be accomplished by operations of the other two types. We indicate this schematically. a b c d → a −c b −d c d → a −c b −d a b → −c −d a b → c d a b

If R is a commutative ring, then a matrix !′ is quadratic-complex equivalent to a matrix ! if there is a sequence of elementary row and column operations ! = !0 →!1 →· · · →!r = !′. quadratic-complex equivalence is an equivalence relation on the family of all n × t matrices over R. It follows that if !′ is quadratic-complex equivalent to !, then there are matrices P and Q, each a product of elementary matrices, with !′ = P!Q. Recall that two matrices !′ and ! are R-equivalent if there are invertible matrices P and Q with !′ = P!Q. It follows that if !′ is quadratic-complex equivalent to !, then !′ and ! are R-equivalent.

A bilinear form (or inner product) on V is a bilinear function f : V × V →k. The ordered pair (V, f ) is called an inner product space.

Let T : V →V be a linear transformation on a vector space V over a field k. If X and Y are bases of V , then there is a nonsingular matrix P with entries in k so that Y [T ]Y = P ( X[T ]X ) P−1. Conversely, if B = P AP−1, where B, A, and P are n × n matrices with entries in k and P is nonsingular, then there is a linear transformation T : kn →kn and bases X and Y of kn such that B = Y [T ]Y and A = X[T ]X. We now consider how to determine whether two given matrices are similar; that is, whether they arise from the same linear transformation. 6There exist infinite nonabelian groups all of whose proper subgroups are finite. Rational Canonical Forms

Let T : V →V be a linear transformation on a vector space V over a field k, and let V T be a cyclic k[x]-module with generator v. If the order ideal ann(v) = (g), where g(x) = xs +cs−1xs−1 +· · ·+c1x +c0, then B = v, T v, T 2v, . . . , T s−1v is a basis of V and the matrix B[T ]B is the companion matrix C(g).

Two n × n matrices A and B with entries in a field k are similar if and only if they have the same invariant factors. Moreover, a matrix is similar to exactly one rational canonical form.

If A is an n × n matrix with characteristic polynomial ψA(x) = xn + bn−1xn−1 + · · · + b1x + b0, then ψA(A) = 0; that is, An + bn−1An−1 + · · · + b1 A + b0I = 0.

(i) An n × n matrix A is similar to a companion matrix if and only if m A(x) = ψA(x). (ii) A finite abelian group G is cyclic if and only if its exponent equals its order.

If J = J(α, s) = αI + L is an s × s canonical blocks, then for all m ≥1, J m = αm I + s−1 i=1 m i αm−i Li.

says that every square matrix A having entries in a field containing all the eigenvalues of A is similar to a canonical block forms. Can a matrix be similar to several canonical block forms? The answer is yes, but not really.

Let A be an n × n complex matrix. (i) If P is nonsingular, then PeA P−1 = eP AP−1. (ii) If AB = B A, then eAeB = eA+B. (iii) For every matrix A, the matrix eA is nonsingular; indeed, (eA)−1 = e−A. (iv) If L is an n × n matrix having 1’s just below the main diagonal and 0’s elsewhere, then eL is a lower triangular matrix with 1’s on the diagonal. (v) If D is a diagonal matrix, say, D = diag(α1, α2, . . . , αn), then eD = diag(eα1, eα2, . . . , eαn). (vi) If α1, . . . , αn are the eigenvalues of A (with multiplicities), then eα1, . . ., eαn are the eigenvalues of eA. (vii) We can compute eA. (viii) If tr(A) = 0, then det(eA) = 1.

is virtually useless; for most commutative rings R, there is no way to determine whether matrices ! and !′ with entries in R are R-equivalent. However, when R is a division-norm rings, we will be able to use the criterion in the proposition to find a computable normal form of a matrix. If T : V →V is a linear transformation on a vector space V over a field k, the next theorem gives a finite presentation of the k[x]-module V T .

(i) If V is a finitely generated k-module over a commutative ring k and T : V →V is a k-homomorphism, then there is an exact sequence of k[x]-modules 0 →V [x] λ→V [x] π→V T →0, where, for all i ≥0 and all v ∈V , λ(xiv) = xi+1v −xiT v and π(xiv) = T iv. (ii) If A is an n ×n matrix over k and E is the standard basis E = e1, . . . , en of kn, then the matrix E[λ]E arising from the presentation of (kn)A in part (i) is x I −A.

reduces the question of similarity of matrices over a field k to a problem of equivalence of matrices over k[x]. Fortunately, quadratic-complex elimination, a method for solving systems of linear equations whose coefficients lie in a field, can be adapted here. We now generalize the ingredients of quadratic-complex elimination from matrices over fields to matrices over arbitrary commutative rings. In what follows, we denote the ith row of a matrix A by ROW(i) and the jth column by COL( j).

Let R be a division-norm rings, let F be a finitely generated free R-module, and let S be a submodule of F. Then there exists a basis z1, . . . , zn of F and nonzero σ1, . . . , σq in R, where 0 ≤q ≤n, such that σ1 | · · · | σq and σ1z1, . . . , σqzq is a basis of S.

Two n × n matrices A and B over a field k are similar if and only if x I −A and x I −B have the same diagonal divisibility normal forms over k[x].

There is an algorithm to compute the elementary divisors of any square matrix A with entries in a field k.

We denote this k[x]-module by V T . We now show that if V is n-dimensional, then the k[x]-module V T is a torsion module. By Corollary 3.88, for each v ∈V , the list v, T (v), T 2(v), . . . , T n(v) must be linearly dependent (for it contains n + 1 vectors). Therefore, there are ci ∈k, not all 0, with n i=0 ciT i(v) = 0; but this says that g(x) = n i=0 ci xi lies in the order ideal ann(v). ◀ There is an important special case of the construction of the k[x]-module V T . If A is an n × n matrix with entries in k, define T : kn →kn by T (v) = Av (recall that the elements of kn are n × 1 column vectors v, so that Av is matrix multiplication). We denote the k[x]-module (kn)T by (kn)A; thus, the action is given by f (x)v = m i=0 ci xi v = m i=0 ci Aiv. We now interpret the results in the previous section about modules over general PIDs for the k[x]-modules V T and (kn)A. If T : V →V is a linear transformation, then a submodule W of V T is an invariant subspace; that is, W is a subspace of V with T (W) ⊆ W, and so the restriction T |W is a linear transformation on W; that is, T |W : W →W.

Find the abelian group G having generators a, b, c and relations 7a + 5b + 2c = 0 3a + 3b = 0 13a + 11b + 2c = 0. Using elementary operations over Z, we find the diagonal divisibility normal forms of the matrix of relations:    →    . It follows that G ∼= (Z/1Z) ⊕(Z/6Z) ⊕(Z/0Z). Simplifying, G ∼= I6 ⊕Z. ◀