Rational Canonical Form

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1 Introduction k[x]-modules Matrix Representation of Cyclic Submodules The Decomposition Theorem May 2014

2 Introduction k[x]-modules Matrix Representation of Cyclic Submodules The Decomposition Theorem Table of Contents Introduction k[x]-modules Matrix Representation of Cyclic Submodules The Decomposition Theorem

3 Introduction k[x]-modules Matrix Representation of Cyclic Submodules The Decomposition Theorem Why do we need? Consider the matrix over R, A = This matrix has characteristic polynomial x 4 + 9x 3 97x x 9226

4 Introduction k[x]-modules Matrix Representation of Cyclic Submodules The Decomposition Theorem Why do we need? Consider the matrix over R, A = This matrix has characteristic polynomial x 4 + 9x 3 97x x 9226 Can not find Jordan Canonical Form for this matrix.

5 Introduction k[x]-modules Matrix Representation of Cyclic Submodules The Decomposition Theorem What is? Recall that a companion matrix for a polynomial f (x) = x n + a n 1 x n a 0 is the matrix of the form: a a a a a n 1

6 Introduction k[x]-modules Matrix Representation of Cyclic Submodules The Decomposition Theorem A matrix in is a matrix of the form C[f n ] C[f n 1 ]... C[f1] Where C[f i ] is a companion matrix for the polynomial f i. Furthermore, f n f n 1... f 1.

7 Introduction k[x]-modules Matrix Representation of Cyclic Submodules The Decomposition Theorem k[x]-modules Definition Recall that a k[x]-module is a module with scalars from the ring k[x] and scalar multiplication defined as follows: Given f (x) k[x], f (x)v = n a i x i v = n a i T i (v) = f (T )(v). i=0 i=0

8 Introduction k[x]-modules Matrix Representation of Cyclic Submodules The Decomposition Theorem k[x]-modules Definition Recall that a k[x]-module is a module with scalars from the ring k[x] and scalar multiplication defined as follows: Given f (x) k[x], f (x)v = n a i x i v = n a i T i (v) = f (T )(v). i=0 We can think of this as the module associated with the linear transformation T i=0

9 Introduction k[x]-modules Matrix Representation of Cyclic Submodules The Decomposition Theorem Definition Given an R-module, M, and m M, the annihilator of m M is: ann(m) = {r R : rm = 0}. Theorem Given a vector space V over a field F and a linear transformation T : V V, the F [x]-module, V T, is a torsion module.

10 Introduction k[x]-modules Matrix Representation of Cyclic Submodules The Decomposition Theorem Definition Given an R-module, M, and m M, the annihilator of m M is: ann(m) = {r R : rm = 0}. Theorem Given a vector space V over a field F and a linear transformation T : V V, the F [x]-module, V T, is a torsion module. Proof. the set {v, T (v), T 2 (v),..., T n (v)} is linearly dependent since it contains n + 1 vectors. g(x) = n a i x i ann(v) i=0

11 Introduction k[x]-modules Matrix Representation of Cyclic Submodules The Decomposition Theorem Definition If M is an R-module, then a submodule N of M, denoted N M is an additive subgroup N of M closed under scalar multiplication. That is, rn N for n N and r R.

12 Introduction k[x]-modules Matrix Representation of Cyclic Submodules The Decomposition Theorem Definition If M is an R-module, then a submodule N of M, denoted N M is an additive subgroup N of M closed under scalar multiplication. That is, rn N for n N and r R. Theorem Given a vector space V over a field F and a linear transformation, T : V V, a submodule W of the F [x]-module V T is a T -invariant subspace. More specifically, T (W ) W.

13 Introduction k[x]-modules Matrix Representation of Cyclic Submodules The Decomposition Theorem The Minimal Polynomial and k[x]-modules Definition The annihilator of a module, M, is: ann(m) = {r R : rm = 0 for all m M} Definition The Minimal Polynomial of a matrix A, denoted m A (x), is the unique monic polynomial of least degree such that m A (A) = 0.

14 Introduction k[x]-modules Matrix Representation of Cyclic Submodules The Decomposition Theorem The Minimal Polynomial and k[x]-modules These two terms are related for k[x]-modules

15 Introduction k[x]-modules Matrix Representation of Cyclic Submodules The Decomposition Theorem The Minimal Polynomial and k[x]-modules These two terms are related for k[x]-modules ann(v T ) ={f (x) F [x] f (x)v = 0 for all v V } ={f (x) F [x] f (T )v = 0 for all v V } ={f (x) F [x] f (T ) = 0} We can use these terms synonymously

16 Introduction k[x]-modules Matrix Representation of Cyclic Submodules The Decomposition Theorem Matrix Representation of Cyclic Submodules Definition Given an R-module, M, and an element m M, the cyclic submodule generated by m is m = {rm : r R}

17 Introduction k[x]-modules Matrix Representation of Cyclic Submodules The Decomposition Theorem Matrix Representation of Cyclic Submodules Definition Given an R-module, M, and an element m M, the cyclic submodule generated by m is m = {rm : r R} Since a submodule, W, of a k[x]-module is T -invariant, we can examine the matrix representation T W

18 Introduction k[x]-modules Matrix Representation of Cyclic Submodules The Decomposition Theorem Matrix Representation of Cyclic Submodules Definition Given an R-module, M, and an element m M, the cyclic submodule generated by m is m = {rm : r R} Since a submodule, W, of a k[x]-module is T -invariant, we can examine the matrix representation T W Let us look at T restricted to cyclic submodules of k[x]-modules

19 Introduction k[x]-modules Matrix Representation of Cyclic Submodules The Decomposition Theorem Theorem Let W = w be a cyclic submodule of the F [x]-module V T and deg(m T W (x)) = n. Then the set {T n 1 (w), T n 2 (w),..., T (w), w} is a basis for W. Proof. By the division algorithm, we can write any polynomial f (x) = m(x)q(x) + r(x) where m(x) is the minimal polynomial of T W with deg=n and deg(r(x)) < n so, for any w 1 W, w 1 =r(x)w =r(t )w =a n 1 T n 1 (w) + a n 2 T n 2 (w) a 0 (w).

20 Introduction k[x]-modules Matrix Representation of Cyclic Submodules The Decomposition Theorem Proof cont. Consider the relation of linear dependence: a n 1 T n 1 (w) + a n 2 T n 2 (w) a 0 (w) = 0 a n 1 T n 1 (w) + a n 2 T n 2 (w) a 0 (w) = p(x)w, deg(p(x)) < deg(m(x))

21 Introduction k[x]-modules Matrix Representation of Cyclic Submodules The Decomposition Theorem Now consider the matrix representation of T W relative to the basis {w, T (w),..., T n 1 (w)}

22 Introduction k[x]-modules Matrix Representation of Cyclic Submodules The Decomposition Theorem Now consider the matrix representation of T W relative to the basis {w, T (w),..., T n 1 (w)} a a a a a n 1

23 Introduction k[x]-modules Matrix Representation of Cyclic Submodules The Decomposition Theorem Primary Decomposition Theorem Let M be a finitely generated torsion module over a principal ideal domain, D, and let ann(m) = u, u = p e 1 1 pe pen n where each p i is prime in D. Then M = M p1 M p2... M pn where M pi = {v V : pi ev = 0}.

24 Introduction k[x]-modules Matrix Representation of Cyclic Submodules The Decomposition Theorem Cyclic Decomposition Theorem Let M be a primary, finitely generated torsion module over a principle ideal domain, R with ann(m) = p e, then M is the direct sum, M = v 1 v 2... v n where ann( v i ) = p e i and the terms in each cyclic decomposition can be arranged such that ann(v 1 ) ann(v 2 )... ann(v n ).

25 Introduction k[x]-modules Matrix Representation of Cyclic Submodules The Decomposition Theorem Therefore, we can write: V T = M p1 M p2... M pn = ( v 1,1 v 1,2... v 1,k1 )... ( v n,1... v n,kn )

26 Introduction k[x]-modules Matrix Representation of Cyclic Submodules The Decomposition Theorem Therefore, we can write: V T = M p1 M p2... M pn = ( v 1,1 v 1,2... v 1,k1 )... ( v n,1... v n,kn ) ann( v i,j ) = p e i,j i

27 Introduction k[x]-modules Matrix Representation of Cyclic Submodules The Decomposition Theorem Therefore, we can write: V T = M p1 M p2... M pn = ( v 1,1 v 1,2... v 1,k1 )... ( v n,1... v n,kn ) ann( v i,j ) = p e i,j i p e i i = p e i,1 i p e i,2 i... p e i,k i i

28 Introduction k[x]-modules Matrix Representation of Cyclic Submodules The Decomposition Theorem The Invariant Factor Decomposition We can rearange these cyclic subspaces into the following groups W 1 = v 1,1 v 2,1... v n,1 W 2 = v 1,2 v 2,2... v n,2. each W i is cyclic with order p e 1,i p e 2,i...p e j,i = d i Each d i is called an invariant factor of V T Notice that since d 1 = p e 1,1 1 p e 2,1 2...p e n,1 n, d 2 = p e 1,2 We can conclude that d n d n 1... d 1 1 p e 2,2 2...p e n,2 n,...

29 Introduction k[x]-modules Matrix Representation of Cyclic Submodules The Decomposition Theorem Example Suppose that W is a torsion module with order p e 1 1 pe 2 2 pe 3 3 W = M p1 M p2 M p3 Suppose that M p1 M p2 M p3 = ( v 1,1 v 1,2 v 1,3 ) ( v 2,1 v 2,2 ) ( v 3,1 ) Then the p e 1 1 = ann(v 1,1) ann(v 1,2 ) ann(v 1,3 ), p e 2 2 = ann(v 2,1 ) ann(v 2,2 ), p e 3 3 = ann(v 3,1). W = ( v 1,1 v 2,1 v 3,1 ) ( v 1,2 v 2,2 ) ( v 1,3 ) d 1 = p e 1,1 1 p e 2,1 2 p e 3,1 3 = p e 1 1 pe 2 2 pe 3 3, d 2 = p e 1,2 1 p e 2,2 2, and d 3 = p e 1,3 1

30 Introduction k[x]-modules Matrix Representation of Cyclic Submodules The Decomposition Theorem Given any matrix, we can realize this matrix as the linear transformation, T, associated with the k[x] module, V T

31 Introduction k[x]-modules Matrix Representation of Cyclic Submodules The Decomposition Theorem Given any matrix, we can realize this matrix as the linear transformation, T, associated with the k[x] module, V T The first invariant factor will be the minimum polynomial

32 Introduction k[x]-modules Matrix Representation of Cyclic Submodules The Decomposition Theorem Given any matrix, we can realize this matrix as the linear transformation, T, associated with the k[x] module, V T The first invariant factor will be the minimum polynomial Each invariant factor will be a factor of the minimum polynomial

33 Introduction k[x]-modules Matrix Representation of Cyclic Submodules The Decomposition Theorem Example Consider the matrix,

34 Introduction k[x]-modules Matrix Representation of Cyclic Submodules The Decomposition Theorem Example Consider the matrix, characteristic polynomial is x 3 + 6x x + 8 = (x + 2) 3

35 Introduction k[x]-modules Matrix Representation of Cyclic Submodules The Decomposition Theorem Example Consider the matrix, characteristic polynomial is x 3 + 6x x + 8 = (x + 2) 3 minimal polynomial is (x + 2) 2 since (A + 2I ) 2 = 0 invariant factors are (x + 2) 2 and x + 2

36 Introduction k[x]-modules Matrix Representation of Cyclic Submodules The Decomposition Theorem Example cont. Therefore, the rational canonical form of this matrix is:

Algebra Homework, Edition 2 9 September 2010

Algebra Homework, Edition 2 9 September 2010 Problem 6. (1) Let I and J be ideals of a commutative ring R with I + J = R. Prove that IJ = I J. (2) Let I, J, and K be ideals of a principal ideal domain.