Chapter 7

THE PRIMARY AND THE CYCLIC DECOMPOSITION THEOREMS

1. The Primary Decomposition Theorem

If the minimal polynomial p_T(x) of a linear operator T on a finite dimensional vector spave V has a maximum of k distint prime polynomial factors, the primary decomposition theorem enables us to decompose V into a direct sum of k T-invariant subspaces on each of which the minimal polynomial of the restriction of the operator T has a simple form, thereby reducing the dimensionality of the problem in studying T through the restrictions of T to these subspaces. The primary decomposition theorem, combined with the cyclic decomposition theorem in the sequel, also leads to a derivation of the important Jordan canonical form of an operator.

Lemma. Let T be a linear operator on a vector space V . Let P₁, P₂, … , P_k be monic polynomials, such that (P_i, P_j) = 1, i ¹ j, and let P(T) = 0, where P = P₁P₂ … P_k . Let V _i = N (P_i(T)). Then:

(i) V _i 's are T-invariant;

(ii) V = Å ₁£ i£ k V _i = V ₁ Å V ₂ Å … Å V _k;

(iii) If T_i = T_|V i and g_i = P ₁£ j£ k, j¹ i P_j, g_i(T_i) is non singular.

Proof: The polynomials g_i , 1 £ i £ k, are relatively prime, i.e., (g₁, g₂, … ,g_k) = 1. [For, if a prime g | g_i for all i, g | P_j for some j. But then g Œ g_j, else g has to divide P_i for some i ¹ j and then P_i and P_j would not be relatively prime, a contradiction]. Hence there exist polynomials h₁, … , h_k such that 1 = S ₁£ i£ k g_ih_i. Then I = S ₁£ i£ k g_i(T)h_i(T) = S ₁£ i£ k E_i, say, where E_i = g_i(T)h_i(T). It follows that E_i² = E_i (i.e. E_i 's are projections), and, E_iE_j = 0, i ¹ j. Putting W _i = R (E_i), we have V = Å ₁£ i£ k W _i = W ₁ Å W ₂ Å … Å W _k. Since E_i 's commute with T, W _i 's are T-invariant. Note that P_i(T)E_i = 0 (as P_i(T)E_i = P_i(T)g_i(T)h_i(T)). Hence W _i Ì V _i. Also, if v Î V _i , then v = E₁v + … + E_iv + … + E_kv = E_iv Î W _i (if j ¹ i, E_j has a factor P_i(T)). It follows that V _i Ì W _i. Hence W _i = V _i. Finally, g_i(T)h_i(T) = E_i is an identity on W _i, and so g_i(T_i) is non-singular as g_i(T_i) = g_i(T_|V i ). This completes the proof. #

Corollary 1. (Primary Decomposition Theorem). Let T be a linear operator on a vector space V over a field F . If p_T = P ₁£ i£ k p_i^ri, where the p_i 's are distinct irreducible monic polynomials, and if W _i = N (p_i^ri(T)), 1 £ i £ k, then:

(i) each W _i is T-invariant;

(ii) V = W ₁ Å … Å W _k;

(iii) if T_i = T_|W i, p_Ti = p_i^ri.

Proof: Only the last assertion is to be verified. Let q_i = p_Ti. Putting q = P ₁£ i£ k q_i, q(T) = q₁(T)q₂(T) … q_k(T) = 0, since for w_i Î W _i, q(T)w_i = q(T)w_i = q(T_i)w_i = (P _j¹ i q_j(T_j))q_i(T_i)w_i = 0. Hence, p_T | q = P ₁£ i£ k q_i. As W _i = N (p_i^ri(T)), q_i | p_i^ri. so that, q | p_T, and being being monic, q = p_T. #

Problem. Find the primary decomposition of F ⁴ relative to each of the following matrix operators:

.

Corollary 2. If E₁, … , E_k are the projections associated with the primary decomposition theorem, then each E_i is a polynomial is T;hence if an operator S commutes with T it commutes with each E_i, i.e., each W _i is invariant under S.

Corollary 3. Let f_T = P₁P₂ … P_k, where P_i 's are pair-wise relatively prime and monic. Then f_Ti = P_i. [P_i 's, T_i 's as in the Lemma.]

Proof: By the Lemma, pooling the bases of V_i 's with f_i º f_Ti , P₁P₂ … P_k = f_T = f₁ … f_k. [Later we will see that every prime factor of the characteristic polynomial is also a factor of the minimal polynomial;and that if q is a factor of p then q(T) is singular]. Since P₂(T₁) … P_k(T₁) is non-singular, by the Lemma, no prime factor of f₁ is a factor of P₂ … P_k. It follows that f₁ | P₁ (by exhausting the prime factors). Similarly, f_i | P_i , 1 £ i £ k. As P f_i = P P_i, f_i = P_i, 1 £ i £ k . #

Proof: Let p_T(x) = P ₁£ i£ k (x-c_i)^mi. Let E_i 's be the projections associated with the primary decomposition V = W ₁ Å W ₂ Å … Å W _k , where W _i = N (( c_iI-T)^di). Then E_i 's are polynomials in T and E_iE_j = d _ijE_i. Hence, D = S ₁£ i£ k c_iE_i is diagonalizable and is a polynomial in T. Hence, N = T – D is a polynomial in T and therefore ND = DN. Since I = Å ₁£ i£ k E_i, N = S ₁£ i£ k (c_iI-T)E_i. Hence if m = max m_i , we have N^m = S ₁£ i£ k (c_iI-T)^mE_i = 0, so that N is nilpotent, and T = D + N. If also T = D₁ + N₁, D₁N₁ = N₁D₁, D₁ is diagonalizable and N₁ is nilpotent, then D₁ and N₁ commute with T (e.g., D₁T = D₁(D₁+N₁) = (D₁+N₁)D₁ = TD₁) and so with D and N. Hence D-D₁ = N₁-N is diagonalizable (both D and D₁ being so and commuting and so being simultaneously diagonalizable) and also is nilpotent as (N–N₁)^d = 0,if d = max {2d₁, 2d₂ : N^d1 = N₁^d1 = 0}. It follows that D-D₁ = 0 implying that D = D₁ and N = N₁. #

Jordan Canonical Form (Assuming the primary decomposition). Let T be nilpotent. As W = T(V ) is T-invariant, let S denote the restriction of T to W . As dim(W ) < dim(V ), by induction on the dimension of space, we can assume that S has a Jordan basis È ₁£ i£ m{w_ij Î W : 1 £ j £ m_i} in W with Tw_i1 = 0, 1 £ i £ m and Tw_ij = w_ij-1 , 1 < j £ m , 1 £ i £ m. Since T(S _i³ 1 S _j³ 1 c_ijw_ij ) = S _i³ 1 S _j>1 c_ijw_ij-1, it is clear that B ₁ = {w_i1: 1 £ i £ m} is a basis of N (S). We can extend B ₁, by choosing B ₂ = {u_k Î V \W :1 £ k £ l}, so that span{B ₁ È B ₂} = N (T). Finally, as W = T(V ), there exist v_i Î V such that Tv_i = w_imi , 1 £ i £ m. It is clear that v_i 's are linearly independent. Noting that T(S _i³ 1 S _j£ mi a_ijT^jv_i + S b_ku_k) = S _i³ 1 S _j<mi a_ij T^j+1v_i, it follows that B = {T^jv_i : 0 £ j £ m_i, 1 £ i £ m} È B ₂ is a Jordan basis for T in V , proving the result for a nilpotent T.

For a general T on V , if f_T(x) = P ₁£ i£ p (x-c_i)ⁿi, where c_i 's are distinct, g_i(x) = f_T(x)/(x-c_i)ⁿi are relatively prime so that for some polynomials h_i(x) there holds 1 = S _i g_i(x)h_i(x). It follows that V = W ₁ Å W ₂ Å … Å W _p, a direct sum, where W _i = g_i (T)h (T)V = N ((T-c_iI)ⁿi), 1 £ i £ p, are T-invariant. As the operator T_i = T-c_iI is nilpotent on W _i, T_i has a Jordan basis B _i in W _i. From this it is clear that B = B ₁ È B ₂ È … È B _p is a Jordan basis for T in V . #

PROBLEMS

1. Let T be the linear operator on R ³, represented in the standard ordered basis by the matrix

.

Show that p_T can be expressed in the form p = p₁p₂, where p₁ and p₂ are monic and irreducible over the field of real numbers. With W _i = N (p_i(T)) and T_i = T|W _i, choose some ordered bases B _i for the subspaces W _i and determine the matrix of T_i in the basis B _i, and hence that of T with respect to the pooled basis B = {B ₁, B ₂}.

2. Find the commuting diagonalizable and nilpotent parts for whichever of the following matrix operators over R ³ it is possible:

.

4. What are the commuting diagonalizable and nilpotent parts of the differentiation operator on P _n, the space of all polynomials of degree less than or equal to n over a field F .

5. If T is a linear operator on a vector space V such that p_T(x) = (x-c₁) ^r1 … (x-c_k)^rk , (c_i 's distinct). Show that W = N ((T-c_iI) ^ri) consists precisely of all those vectors v Î V for each of which there exists a positive integer m such that (T-c_iI)^mv = 0, and that dim W _i = d_i, where f_T(x) = (x-c₁) ^d1 … (x-c_k) ^dk.

6. Let T be a triangulable linear operator on a finite dimensional vector space V over a field F and let D be the diagonalizable part of T. Prove that if g is a polynomial over F, g(T) is triangulabel and the diagonalizable part of g(T) is g(D).

8. Characterize all linear operators T, on a finite-dimensional vector space V over a field F , that commute with every diagonalizable linear operator on V . Is any of them not a scalar multiple of the identity operator?

9. Let A Î F ⁿ´ n, and define T_A(B) = AB-BA, B Î F ⁿ´ n. Prove that A is triangulable, diagonalizable, or, nilpotent iff T is so.

10. In which of the 2´ 2, 3´ 3, 4´ 4 and 5´ 5 cases do there exist non-similar nilpotent matrices having the same minimal polynomial.

11. If V = W ₁ Å … Å W _k is the primary decomposition for T and W a T-invariant subspace, show that W = (W Ç W ₁) Å … Å (W Ç W _k).

12. If D is an n´ n diagonal matrix and N an n´ n upper triangular matrix, is it true that the diagonalizable and the nilpotent parts of the matrix A = D + N are D and N, respectively? Would you make a concession if the diagonal entries in D were distinct?

13. If T is a linear operator on V with p_T = p^m, where p is irreducible over the scalar field F , prove that the T-annihilator s^T_u of 0 ¹ u Î V is of the form p^k, 1 £ k £ m;and moreover, that any such p^k is the T-annihilator of some 0 ¹ u Î V .

15. Show that all nilpotent linear operator on an n-dimensional vector space V have identical characteristic polynomials. Do they have the same minimal polynomials too?

2. T-Cyclic Subspaces and Vectors

A subspace W of a vector space V is called a T-cyclic subspace if there is a w Î V such that W = Z (w;T) = <w,Tw,T²w, … > (<S> denoting the span of the set S) and we say that W has a T-cyclic vector w. If v is a T-cyclic vector for V , we simply say that v is a T-cyclic vector. Clearly, Z (u;T) = {P(T)u: P a polynomial}.

Let s^T_v(x) denote the T-annihilator of v, i.e., the monic polynomial of least degree such that s^T_v(T)v = 0. Let s^T_v be of degree k so that s^T_v(x) = a₀ + a₁x + … + a_k-1x^k-1 + x^k. Then B = {v, Tv, … , T^k-1v} is a basis of Z (v;T) and the matrix of T_|Z (v;T) with respect to the basis B is

The matrix C is called the companion matrix of the monic polynomial P(x) = a₀ + a₁x + … + a_k-1x^k-1 + x^k. Obviously, p_T|Z (v;T) = f_T|Z (v;T) = s^T_v = p_C = f_C = s^T_v. Also note that given any monic polynomial P(x) = a₀ + a₁x + … + a_k-1x^k-1 + x^k, C is the matrix, w.r.t the standard basis {e₁, … ,e_k} of F ^k, of the linear operator S defined by Se_i = e_I+1, i = 1, … , k-1, Se_k = -(a₀e₁ + a₁e₂ + … + a_k-1e_k). It follows that the characteristic and minimal polynomials of a companion matrix C (of any monic polynomial P) are identical.

Lemma 1. If Q is a polynomial Q(T)Z (v;T) = Z (Q(T)v;T).

Proof: Q(T)Z (v;T) = Q(T){P(T)v : P Î F [x]} = {Q(T)P(T)v : P Î F [x]} = {P(T)Q(T)v : P Î F [x]} = Z (Q(T)v;T). #

Proof: There exist polynomials Q and S such that (P, s^T_v) = PQ + s^T_vS. Let R(Pv) = 0. Þ R(P,s^T_v)v = 0. Þ s^T_v | R(P, s^T_v). Þ s^T_v ¤ (P, s^T_v) ½ R. Finally, (s^T_v ¤ (P, s^T_v)(Pv) = (P/(P, s^T_v))s^T_vv = 0. #

PROBLEMS

1. What linear operators on F ¹ possess a cyclic vector? Show that a linear operator on F ² possesses a cyclic vector iff it is not a scalar multiple of the identity operator. Is the result true in F ⁿ for n > 2.

on F ³ possess a cyclic vector? Show that the cyclic subspace generated by the vector (a,b,c)¢ is two dimensional iff a+b+c ¹ 0 and a = b = c does not hold.

4. Let A denote the matrix operator on C ³ given by

Show that W = span {(1,1,1)¢ } is A-invariant. Find the A-conductor of x = (1,2,1)¢ into W . Is it different from the A-annihilator of x?

6. Prove: a nilpotent operator N on an n-dimensional vector space V over a field F has a cyclic vector iff N ¹ 0.

7. Prove that N is a nilpotent operator on an n-dimensional vector space V and N ¹ 0, iff there is a basis b of V such that

[N]b = ,

where I is the (n-1)´ (n-1) identity matrix.

9. Let V have a basis {v₁, … , v_n} consisting of eigenvectors of a linear operator T possessing a cyclic vector. If v Î V , prove that Z (v;T) = V iff v = c₁v₁ + … + c_nv_n, where c_i ¹ 0, 1 £ i £ n.

10. Let U and T be linear operators on a finite-dimensional vector space V . If T has a cyclic vector v and Uv = P(T)v, where P is a polynomial, prove that U = P(T) iff U commutes with T.

3. The Cyclic Decomposition Theorem

A subspace W is called T-admissible if W is T-invariant and if for every polynomial g and every v Î V , g(T)v Î W implies that there exists a vector w Î W such that g(T)v = g(T)w.

Lemma 1. Let W ₀ be T-invariant and let u Î V be such that s^T_u,W 0(x) is of maximal degree. Let W = W ₀ + Z (u;T). Then, W is T-invariant, and, for any v Î V if s^T_v,W (T)v = w₀ + P(T)u, (w₀ Î W ₀, P Î F [x]) then s^T_v,W (x) ½ P(x).

Proof: Let P(x) = s^T_v,W (x)Q(x) + R(x), where deg R < deg s^T_v,W (x). Let w = v - Q(T)u. Then s^T_w,W (x) = s^T_v,W (x), (as u Î W ) and s^T_w,W (T)w = w₀ + R(T)u.

Since s^T_w,W ½ s^T_w,W 0, we can write s^T_w,W 0(x) = s^T_w,W (x)S(x), where S is a monic polynomial. Then S(T)R(T)u Î W ₀. It follows that s^T_u,W 0 ½ SR. Therefore, if R ¹ 0, then deg (SR) ³ deg (s^T_u,W 0) ³ deg s^T_w,W 0 = deg (Ss^T_w,W ) = deg (Ss^T_v,W ), implying that deg R ³ deg s^T_v,W , a contradiction. Hence R = 0, and consequently, s^T_v,W ½ P. #

Lemma 2. If W ₀ is T-admissible and s^T_u,W is of maximal degree, then W = W ₀ + Z (u;T) is T-admissible.

Proof: By Lemma 1, s^T_v,W (T)v = w₀ + s^T_v,W (T)Q(T)u, for some w₀ Î W ₀ and polynomial Q. Þ w₀ = s^T_v,W (T)(v-Q(T)u) = s^T_v,W (T)w₀¹, for some w₀¹ Î W ₀ (by T-admissibility of W ₀). Þ s^T_v,W (T)v = s^T_v,W (T)(w₀¹ + Q(T)u) = s^T_v,W (T)w, say, where w = w₀¹ + Q(T)u Î W . Now, if P₁(T)v Î W Þ P₁(x) = s^T_v,W (x)Q₁(x) Þ P₁(T)v = Q₁(T)( s^T_v,W (T)v) = Q₁(T)( s^T_v,W (T)w) = P₁(T)w, w Î W, proving that W is T-admissible. #

Lemma 3. If W ₀ is T-admissible and W = W ₀ + Z (u;T), there exists a v Î V such that W = W ₀ Å Z (v;T) and s^T_v,W 0 = s^T_v = s^T_u,W 0.

Proof: Let s^T_u,W 0(T)u = s^T_u,W 0(T)w₀, w₀ Î W ₀. Put v = u – w₀. Then, s^T_u,W 0 = s^T_v,W 0 = s^T_v, and W ₀ + Z (u;T) = W ₀ + Z(v;T). Since W ₀ Ç Z (v;T) = {0}, Þ W = W ₀ Å Z (v;T). #

Theorem (Cyclic Decomposition). Let W ₀ be a proper T-admissible subspace of a finite dimensional vector space V . Then there exist non-zero vectors u₁, u₂, … , u_r Î V , with the respective T-annihilators p₁, p₂, … , p_r such that p_i+1 | p_i , i = 1, 2, … , r-1 and V = W ₀ Å Z (u₁;T) Å … Å Z (u_r;T). Moreover, the number r and the polynomials p₁, p₂, … , p_r are invariants of the decomposition (i.e., they depend only on V , W ₀ and T and not on u_i 's).

Proof: Applying Lemmas 2 and 3 successively, there exist u₁, u₂, … such that W ₁ = W ₀ Å Z (u₁;T), W ₂ = W ₁ Å Z (u₂;T), … , W _k = W _k-1 Å Z (u_k;T), … , are T-admissible and for some r, eventually culminate in V = W _r = W _r-1 Å Z (u_r;T). Let p_j = s^T_uj,W j-1 = s^T_uj. Since 0 = p_i+1(T)u_i+1 = (0 + p₁(T)u₁ + … + p_i-1(T)u_i-1 ) + p_i(T)u_i, Lemma 1, applied to u_i+1 , W _i-1 and W _i , gives p_i+1(x) | p_i(x), i = 1, 2, ... r-1.

To prove the invariance, finally, let also V = W ₀ Å Z (v₁;T) Å … Å Z (v_s;T) where v_i ‘s are non-zero and q_i = s^T_vi and put X _k = X _k-1 Å Z (v_k;T), 1 £ k £ s. As dim W ₀ < dim V it follows that s ³ 1. Then p₁(T)V = p₁(t)W ₀ = p₁(T)W ₀ Å Z (p₁(T)v₁;T) Å … Å Z (p₁(T)v_s;T). It follows that p₁(T)v₁ = 0 and hence that q₁ | p₁. By symmetry, p₁ | q₁, so that p₁ = q₁. Since dim Z (u;T) = deg (s^T_u ) it follows that s ³ 2 (as dim X ₁ = dim W ₁ < dim V ).

Next, p₂(T)V = p₂(T)W ₀ Å Z (p₂(T)u₁;T) = p₂(T)W ₀ Å Z (p₂(T)v₁;T) Å Z (p₂(T)v₂, T) Å … . Hence, p₂(T)v₂ = 0 so that q₂ | p₂. By symmetry p₂ | q₂ and hence p₂ = q₂. It follows that s ³ 3 (as dim X ₂ = dim W ₂ < dim V ). Continuing in this fashion, we get q_i = p_i for all possible i and that s = r. This completes the proof. #

By the cyclic decomposition of V relative to T, in the sequel, we shall mean the above cyclic decomposition of V relative to T when W ₀ = {0}.

Problem. Taking V = F ⁿ and W ₀ = span {(1, 0, 0, … , 0)¢ } find a cyclic decomposition of V relative to each of (i) T = I, (ii) T = 0, and (iii) T = diag (1, 2, ... , n).

Corollary 1. Let W ₀ be T-invariant. Then W ₀ has a complementary T-invariant subspace (i.e., a W such that TW Ì W and W ₀ Å W = V ) iff W ₀ is T-admissible.

Proof: If W ₀ is T-admissible, by the cyclic decomposition theorem, W = Å ₁£ i£ r Z (u_i;T) is a required complementary T-invariant subspace. Conversely, let W be T-invariant and V = W ₀ Å W . Let v = w₀ + w Î V and let P(T)v = P(T)w₀ + P(T)w Î W ₀, (w₀ Î W ₀, w Î W ). Then by T-invariance and linear independence of W ₀ and W , P(T)w = 0 and so P(T)v = P(T)w₀, i.e., W ₀ is T-admissible. #

Corollary 2.(Generalized Cayley-Hamilton Theorem). If p = p₁^m1 p₂^m2 … p_k^mr and V = W ₁ Å … Å W _r correspond to the primary decomposition of T, then f_T = p₁^d1 p₂^d2 … p_r^dr, where m_i £ d_i = (dim W _i)/(deg p_i), 1 £ i £ r.

Proof: With W ₀ = {0} in the cyclic decomposition theorem, s^T_u1 = p_T. As s^T_ui+1 | s^T_ui, i = 1, … , r-1, Þ f_T = P ₁£ i£ r s^T_ui = P ₁£ i£ r p_i^di for some d_i ³ m_i , i = 1, … , r. Since p_i^mi = p_Ti , where T_i = T|W _i , applying the result obtained so far to W _i and T_i Þ f_Ti = p_i^mi , where m_i = (dim W _i)/(deg p_i ). But f_T = P ₁£ i£ r f_Ti = P ₁£ i£ r p_i^mi = P ₁£ i£ r p_i^di Þ (as p_i 's are irreducible (prime) polynomials) that d_i = m_i. #

Aliter: p_Ti = p_i^mi and in the cyclic decomposition of W _i with W ₀ = {0}, we have s^Ti_u1 = p_Ti . As s^Ti_uj+1 | s^Ti_uj for all j. It follows that f_Ti = P _j s^Ti_uj = p_i^di where d_i´ (deg p_i) = dim W _i, from which we have f_T = P ₁£ i£ r p_i^di , where d_i = (dim W _i)/(deg p_i). #

Corollary 3. P_T = f_T iff V has a T-cyclic vector u.

Proof: Apply cyclic decomposition theorem with W ₀ = {0}. If f_T = p_T (= p₁) Þ r = 1 and V = {0} Å Z(u₁;T) = Z(u;T), say. Conversely, if V = Z (u;T) = {0} Å Z (u;T), by the same theorem p_T = s^T_u Þ deg p_T = deg s^T_u = dim Z (u;T) = dim V Þ p_T = f_T . #

Proof: Suppose every T-invariant subspace has a complementary T-invariant subspace. Assume, on the contrary, that p_T = p²q, where p is irreducible. Choose u Î V such that v = p(T)q(T)u ¹ 0. Then there is a T-invariant subspace W such that: W = Z (v;T) Å W . We can write u = Q(T)v + w, where Q is a polynomial and w Î W . Then, v = p(T)q(T)u = p(T)q(T)w = 0, a contradiction.

Conversely, let p_T be a product of distinct irreducible factors, p_T = p₁p₂ … p_r , say. Consider the corresponding primary decomposition : V = W ₁ Å W ₂ Å … Å W _r, W _i = N (p_i(T)). Let W be a T-invariant subspace of V and write s(x) = s^T_u,W (x). Then, s(T)u =s(T)w₁ + s(T)w₂ + … + s(T)w_r Î W . Letting E_i 's denote the projections associated with the primary decomposition, W ' E_is(T)u = E_is(T)w_i = s(T)E_iw_i = s(T)w_i.

Suppose w_i Ï W . If p_i | s, then, of course, s(T)w_i = 0; but if p_i Œ s, using the existence of the polynomials P_i and S such that p_iP_i + sS = 1, we have W ' S(T)s(T)w_i = w_i - P_i(T)p_i(T)w₁ = w_i, a contradiction. Hence, if w_i Ï W , then p_i | s and therefore s(T)w_i = 0. Putting w as the sum of w_i 's belonging to W , s(T)u = s(T)w, showing that W is T-admissible. Hence, by the cyclic decomposition theorem, W possesses a complementary T-invariant subspace. This completes the proof. #

The linear operators T for which p_T is a product of distinct irreducible factors, are diagonalizable with respect to any super field of the field F under consideration in which p_T factors completely into a product of linear factors. Such operators are called semi-simple, that is next to simple (i.e., diagonalizable). It follows that if F is algebraically closed, the semi-simple operators are precisely the ones that are diagonalizable. It is also clear that a matrix A over a field F is semi-simple iff A is diagonalizable over G , where G is a super field of F in which f_T splits (i.e., is a product of linear factors).

PROBLEMS

1. Let T be a linear operator on a finite dimensional vector space V . If f_T = p_T, show that a linear operator U on V commutes with T iff it is a polynomial in T.

2. Let T be the linear operator on F ² which is represented in the standard ordered basis by the matrix

.

Let u = (0,1)¢ and v = (1,0)¢ . Show that F ² = Z (u;T) ¹ Z (v;T) and that there is no non-zero vector w Î F ² with Z (w;T) disjoint from Z (v;T). Does T possess a proper T-invariant subspace? Does it possess a proper T-admissible subspace?

3. Let T be any linear operator on the finite-dimensional vector space V . Prove that the following statements are equivalent: (a) N (T) is T-admissible; (b) R (T) is T-admissible; (c) V = N (T) Å R(T); (d) if W is T-invariant and V = N (T) Å W , then W = R (T); and, (e) if U is T-invariant and V = R (T) Å U , then U = N (T).

4. Let T be a linear operator on a finite dimensional vector space V over a field F . Prove that if c Î F , S = T-cI and W is a T-invariant subspace of V , then W is T-admissible iff W is S-admissible.

5. If W ₁, … , W _k are T-invariant subspaces, V = W ₁ Å … Å W _k, and P is a polynomial, verify that P(T)V = P(T)W ₁ Å … Å P(T)W _k.

6. If u, v Î V have the same T-annihilators verify that for any polynomial P, also P(T)u and P(T)v have the same T-annihilator. Now, let us look for a polynomial Q such that for whatever u and v in V , whenever Q(T)u and Q(T)v have the same T-annihilator also u and v have the same T-annihilator. Show that such a non-constant polynomial Q exists. Characterise all such Q's.

7. Let T be the linear operator on F ⁴ which is represented in the standard ordered basis by the matrix

.

Find all the possible T-conductors of elements of F ⁴ into N (T-cI).

12. Let F Ì G be fields and P an irreducible polynomial over F . Prove that P is a product of linear factors over G iff P is a product of distinct linear factors over G . Deduce that two semisimple linear operators T and S over a finite dimensional vector space V over F are similar iff f_T = f_S.

13. Let T be a linear operator on R ², represented in the standard ordered basis by one of the matrices

.

Verify that the T-invariant subspaces N (I-T) and R (I-T) do not possess complementary T-invariant subspaces. Generalize.

4. The Rational Form

Theorem. Let T be a linear operator on a finite dimensional vector space V . Then there exists a basis b of V such that [T]b is in the rational form. Moreover, the rational form of T is unique (i.e., if also [T]g is in the rational form, then [T]b = [T]g ).

Proof: Using cyclic decomposition theorem with W ₀ = 0, we have V = Z (u₁;T) Å Z ( u₂;T) Å … Å Z(u_r;T)where s^T_ui | s^T_ui+1, i = 1, ... , r-1. If deg s^T_ui = d_i , and p_i = s^T_ui, then [T]b = diag (C₁,C₂, … ,C_r)where b = {b ₁,b ₂, … ,b _r} where b _i = {u_i, Tu_i, … ,T^di-1u_i} and c_i is the companion matrix of p_i. For unicity, if q_i+1 | q_i, i = 1, ... , s-1, [T]g = diag (E₁,E₂, ... ,E_s), (E_i companion matrix of q_i) and g = {g ₁, g ₂, … ,g _s} is a compatible de-pooling with g _i = {v_i1, v_i2, … , v_idi}, Þ that with v_i = v_i1, i = 1, ... , s, V = Z (v₁;T) Å Z (v₂;T) S … Å Z (v_s;T) with s^T_vi | s^T_vi+1, i = 1, ... , s-1. Hence, by cyclic decomposition theorem, s = r, s^T_vi = s^T_ui, and therefore [T]g = [T]b . #

PROBLEMS

.

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3. Let Q denote the rational field and T denote the linear operator on Q ³, represented in the standard ordered basis by

.

Show that W = {(c,0,c)¢ : c Î Q } is T-admissible and determine the associated cyclic decomposition and the invariant factors.

.

5. If F is a subfield of C , find the possible rational form/s of the linear operator T on F ⁴ which is represented in the standard ordered basis by the matrix

,

6. Show that, if n £ 3, any two n´ n matrices over a field F are similar iff they have the same characteristic polynomial

7. Let F Ì G be two fields and let A be a matrix over F . Verify that the rational form of A, viewed as a matrix over G, is exactly the same as the rational form of A over F . Deduce that the minimal and the characteristic polynomials p_A and f_A of A are independent of whether A is regrded as a matrix over F or over the field G .

8. If F Ì G are two fields, prove that there does exist a matrix over G which is not a matrix over F but is such that its rational form is a matrix over F .

9. If F is a subfield of an algebraically closed field G and all the eigenvalues of an n´ n matrix A over G lie in F , prove that A over G is similar to a matrix B over F . Verify that the result is not necessarily true if the coefficients of the minimal polynomial of A, and not its eigenvalues, lie in F .

10. For any linear operator T on a finite-dimensional vector space V , show that there exists a vector u (called a separating vector for the algebra of polynomials in T) such that for any polynomial P, P(T)u = 0 implies that P(T) = 0.

11. Let T be a linear operator on a finite dimensional vector space V . Prove that u is a separating vector for the algebra

12. If A and B are n´ n real matrices such that p_A = p_B = x² +1, prove that A and B are similar.

13. If c Î F does not possess a square root in the field F , and A is an n´ n matrix over F such that A² = cI, prove that n is even and that A is similar to the block partitioned matrix

,

I being the m´ m identity matrix with m = n/2.

14. If T is a linear operator on a finite-dimensional vector space V , no non-trivial T-invariant subspace in V possesses a complementary T-invariant subspace iff f_T equals p_T and, moreover, is a power of a polynomial irreducible over F .

16. Prove that a linear operator T on a finite dimensional vector space V has a cyclic vector iff each linear operator that commutes with T is a polynomial in T.

17. If T is a linear operator on a finite-dimensional vector space V over a field F , prove that every non-zero vector in V is a cyclic vector for T iff f_T is irreducible over F .

18. Let A Î R ⁿ´ n. If A as an operator on R ⁿ possesses no proper invariant subspaces, show that p_A(x) = x² + 2bx + c², with |b| < c and b, c Î R . Deduce that A is diagonalizable with respect to C .

5. The Invariant Factors

The polynomials p₁(x), p₂(x), … , p_r(x), associated with the rational form of T are called the invariants factors of T. If [T]b is in rational form with p_i 's denoting the invariant factors it is clear that the Smith normal form of l I-[T] is given by diag [1,1, … ,1,p_r(l ),p_r-1(l ), … ,p₁(l )]. Since the Smith normal forms of l I-[T]b and l I-[T]g for any two bases b and g of V coincide, the invariant factors, as well as the rational form of T could be determined by finding the invariant factors of the l -matrix l I-[T] , where b is any ordered basis of V .

PROBLEMS

2. If F Ì G are fields and A and B are matrices over F that are similar over G , prove that they are also similar over F .

3. Prove that every n´ n l -matrix is l -row equivalent to an upper triangular l -matrix.

4. Prove that A Î F ⁿ´ n has a cyclic vector if and only if all minors of xI-A of order n are relatively prime.

5. Let T be a linear operator on an n-dimensional vector space V over a field F and let q₁(l ), … , q_n(l ) denote the diagonal entries of the Smith normal form of l I-T. Prove that q₁(l ) ¹ 1 iff T is scalar.

6. Find the rational form of a diagonal matrix with distinct eigenvalues c_i 's of multiplicities m_i, 1 £ i £ r?

7. Find the invariant factors and the rational form of the following matrix and determine if the matrix is diagonalizable over R :

.

6. The Jordan Canonical Form

An r´ r Jordan block is an r´ r matrix of type

,

in which all diagonal entries are the same scalar l , all sub diagonal entries are 1 and the remaining entries are 0. A 1´ 1 jordan block is simply the matrix [l ]. We call J(l ) an r-th order Jordan block corresponding to the eigen-value l . The transpose of J(l ) in which instead of the sub-diagonal entries being 1, the super diagonal entries are 1, is also called a Jordan block. However, presently, we will adhere to the first definition as given above.

A block diagonal matrix, diag [J₁(l ₁),J₂(l ₂), … ,J_r(l _r)] in which J_i(l _i) 's are Jordan blocks of certain sizes n_i´ n_i, is said to be a matrix in Jordan canonical form.

Theorem. If T is triangulable (i.e., p_T = p(x-c_i), a product of linear factors) there exists a basis b of V such that [T]b is in Jordan canonical form. Moreover, the Jordan canonical form is unique up to a permutation of the Jordan blocks.

Proof: Consider the primary decomposition V = W ₁ Å W ₂ Å … Å W _r, of V with respect to T, with T_i = T_|W i , p_Ti = (x-c_i)^di . Let S_i = T –c_iI_i , I_i being the identity matrix of appropriate size. Then p_Si = x^di and with W ₀ = {0}, the invariant factors of S_i on W _i are of the form x^k . Then the rational form of S_i w.r.t. an appropriate basis b _i is [S_i]b _i = diag [J_1i(0), J_2i(0), … ,J_kii(0)], where the blocks J_ji(0) are Jordan blocks corresponding to l = 0. Here dim J_ji(0) > dim J_j+1i(0). Hence [T_i]b _i = [S_i+c_iI_i] = diag [J_1i(c_i), J_2i(c_i), … ,J_kii(c_i)]. Hence, with b = {b ₁,b ₂, … ,b _r}, the pooled basis of V , [T]b = diag [J₁₁(c₁), J₂₁(c₁), … ,J_k11(c₁), ... , J_1r(c_r), J_2r(c_r), … ,J_krr(c_r)], a Jordan canonical form. Let g be another basis s.t. [T]g is in Jordan canonical form. Since the diagonal entries of Jordan blocks are precisely the eigenvalues of T which are c₁, c₂, … , c_r, without loss of generality we can assume the Jordan blocks permuted so that dim J_ji ³ dim J_j+1i and [T]g = diag [J₁₁(c₁), J₂₁(c₁), … ,J_m11(c₁), ... , J_1r(c_r), J_2r(c_r), … ,J_mrr(c_r)], Let g = {g ₁, g ₂, … , g _r} be a compatible partitioning where g _I ’s constitute basis members corresponding to Jordan blocks corresponding to eigen value c_i.

It is clear that W _i = N((T-c_iI)^di) = span {b _i}, as [(T-c_iI)^di]g = diag [X₁,X₂, … ,X_i-1,0,X_I+1, … ,X_r], X_j 's are non-singular matrices and 0 is the zero matrix associated to the location of J-blocks corresponding to eigenvalue c_i. Hence, with T_i = T_|W i, [T_i –c_i I]g _i = diag [J_1i(0),J_2i(0), ... ,J_mii(0)], which is in rational form.

Similar consideration w.r.t. basis b leads to [T_i-c_iI]b _i = diag [J_1i(0),J_2i(0), ... ,J_kii(0)], a rational form. By the uniqueness of the rational form. It therefore follows that k_i = m_i, dim J_ji(0) ’s in both [T_i-c_iI]b _i = diag [J_1i(0),J_2i(0), ... ,J_kii(0)] and [T_i –c_i I]g _i = diag [J_1i(0),J_2i(0), ... ,J_mii(0)] are the same for all i and j's. It follows therefore that [T]g = [T]b , after at most a permutation of Jordan blocks. #

PROBLEMS

2. Prove that two 3´ 3 matrices are similar over C iff their characteristic polynomials are the same and their minimal polynomials are also the same.

3. Two real matrices A and B are similar over R iff they are similar over C .

4. If an n´ n real matrix A is diagonalizable over C , then it is triangulable over R iff it is diagonalizable over R .

5. Let A be a 10´ 10 complex matrix. If r(A) = 7 and r(A³) = 2, show that r(A²) = 4.

6. Is the following data consistent for a 10´ 10 complex A: r(A) = 7, r(A²) = 4, r(A³) = 3, and, r(A⁴ ) = 1?

7. Let A be an n´ n complex matrix. Let for some k, r(A^k) = p, and, r(A^k+2) = p-1. Prove that r(A^k+1) = p-1.

7. A Rank Based Determination of the Jordan Canonical Form

Let n_k = n_k(l ) denote the number of k´ k Jordan blocks associated with an eigenvalue l in the JCF of a triangulable matrix A. If J is a p´ p Jordan block with eigenvalue zero, clearly r(J) = p-1, r(J²) = p-2, … , r(J^k) = p-k, … , r(J^p-1) = 1, and r(J^p) = r(J^p+1 ) = … = 0. Hence, if J(l ) is a p´ p Jordan block with eigenvalue l , we have

p - r((J(l )^k) =

Summing this relation for all Jordan blocks we have: n-r(A^k) = S _k£ p kn_p(0)+S _k>p pn_p(0) = S _i£ k in_i(0)+kS _I>k n_i(0), k ³ 0. Note that for k = 0, r(A^k) = r(I) = n. It follows that for k ³ 0, r(A^k) - r(A^k-1) = n_k+1(0) + n_k+2(0) + n_k+3(0) + …, so that d ²(r(A^k)) º r(A^k+1) - 2r(A^k) + r(A^k-1) = n_k(0), k ³ 1. It follows that for each eigenvalue l of A, the number n_k(l ) of k´ k Jordan blocks corresponding to l can be determined by the formula: n_k(l ) = d ²(r((l I-A)^k), k ³ 1.

THEOREM. Let {r_k}_k³ 0 be a non-incereasing sequence of non-negative integers. Then there exists a linear operator T on a vector space V of dimension n = r₀ such that r_k = r(T^k), k ³ 0 iff {r_k} is convex, i.e., d ²(r_k) º r_k+1-2r_k+r_k-1 ³ 0, k ³ 1).

PROOF: The necessity part follows from the formula s ²(r(T^k)) = n_k(0), k ³ 1. To prove the sufficiency, it is enough to construct an r´ r matrix J in Jordan canonical form for which r(J^k) = r_k, k ³ 0. For this we choose the number n_k = n_k(0) of k´ k Jordan blocks corresponding to the eigenvalue 0 as n_k = d ²(r_k) ³ 0, k ³ 1. It follows that n₁ + 2n₂ + … + mn_m = (r₀ -2r₁ +r₂ ) + 2(r₁ -2r₂ +r₃ ) + 3(r₂ -2r₃ +r₄) + … + m(r_m-1 -2r_m +r_m+1) = r₀ + S ₂£ k£ m-1 [(k-1)-2k+(k+1)]r_k -(m+1)r_m + mr_m+1 = r₀ – r_m, where m is smallest such that r_m = r_m+1. Thus r_m has to be the total size of the Jordan blocks corresponding to the eigenvalues that are non-zero. With the matrix J constructed as above, it is clear that r(J^m) = r(J^m+1) = r_m. Moreover, for k < m, we have r(J^k) = n_k+1 + 2n_k+2 + … + (m-k)n_m + r_m = d ²(r_k+1) + 2d ²(r_k+2) + … + (m-k) d ²(r_m) + r_m = r_k - (m-k+1)r_m + (m-k)r_m+1 + r_m = r_k, proving the result. #

PROBLEMS

1. If N₁ and N₂ are n´ n nilpotent matrices and n £ 3, show that N₁ and N₂ are similar if and only if they have the same minimal polynomial. Is the result true for n´ n matrices with n ³ 4?

3. Given that T is a linear operator on a vector space V over a field F and that f_T(x) = x⁹ -3x⁷ +3x⁵ –x³: (a) prove that T is triangulable, (b) find the dimension of V , (c) prove that each of the operators T, T+I and T-I is singular, (d) show that there are precisely twenty seven possibilities for the minimal polynomial p_T(x) and that each of these is associated with a unique Jordan form of T, and, (e) find the Jordan form of T if p_T(x) equals: (i) x³ - x, (ii) x⁴ – x², (iii) x⁵ – x³, and, (iv) x⁷ - 2x⁵ + x³.

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7. If A is an n´ n complex matrix show that trace A = lim_x® ¥ x ln [xⁿ/f_A(x)].

8. The following 3´ 3 complex matrices A satisfy A³ + I = 0 and are mutually dissimilar:

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Are there any more such matrices? What is the total number of such dissimilar matrices? What is the total number of dissimilar n´ n complex matrices A such that Aⁿ + I = 0?

9. Let n ³ 2 and 1 £ k < n be integers, and let N be an nxn matrix over a field F such that Nⁿ = 0 but N^k ¹ 0. Prove that if m ³ n/k, N has no m-th root, i.e., that there is no n´ n matrix A such that A^m = N.

10. Let N₁ and N₂ be n´ n nilpotent matrices, over a field F , of equal rank and having the same minimal polynomial. If n £ 6, prove that N₁ and N₂ are similar. Show that this is not true for n > 6.

11. Let T and S be triangulable linear operators, on a finite dimensional vector space V over a field F , having the same characteristic polynomial and the same minimal polynomial. If also for each c Î F , (T-cI) and (S-cI) have the same nullity and if each eigenvalue of A has algebraic multiplicity less than 7, prove that A and B are similar.

14. If A is triangulable over a field G , show that A and A¢ have the same Jordan form in G . Deduce that A and A¢ over any field F are similar.

15. If A is similar to B and A¢ = P(A), where P is a polynomial, is it true that B¢ = P(B)?

16. If N is n´ n nilpotent matrix over a field F of characteristic zero, p and q are positive integers and a = p/q, prove that the truncated binomial series expansion T below satisfies T^q = (I+N)^p:

17. Prove that every non-singular matrix A over an algebraically closed field G of characteristic zero, possesses an n-th root.

18. Show that iff c ¹ 0, the Jordan form of the matrix is: J(l ) = .

19. Which of the following statements is/are true? (a) The subset of R ⁿ´ n of matrices that are diagonalizable over R is dense in R ⁿ´ n, (b) The subset of R ⁿ´ n of matrices that are diagonlizable over C is dense in R ⁿ´ n, (c) The subset of C ⁿ´ n of matrices that are diagonalizable through a real matrix (i.e., C^-1AC = D, C real) is dense in C ⁿ´ n , and, (d) The subset of C ⁿ´ n of matrices that are diagonlaizable over C is dense in C ⁿ´ n.

20. If A is a complex n´ n matrix, r (A) is its spectral radius and e > 0, there exists a matrix norm P × P such that P AP < r (A)+e .

21. Construct a linear operator T on some vector space V over a field F , of characteristic different from 2, such that p_T(x) = (x-1)²(x+1)² and f_T(x) = (x-1)³(x+1)⁴. Describe the primary decomposition of the vector space under T and determine the projections on the primary components. Find a basis of V in which the matrix of T is in the Jordan form. Also find an explicit direct sum decomposition of the space into T-cyclic subspaces and write down the rational form of T and the associated invariant factors. Repeat the exercise with a field of characteristic 2.

22. Let A denote the matrix operator on R ⁸ given by

.

(a) Show that the diagonal entries of the Smith normal form of xI-A are: 1, 1, 1, 1, 1, 1, x²(x+1), and, x²(x+1)³. (b) Deduce the characteristic polynomial, the minimal polynomial and the invariant factors of A associated with its rational form. (c) Find the rational form of A. Find the Jordan form for each diagonal block in the rational form of A. (d) Find the cyclic decomposition of R ⁸ under T relative to W ₀ = {0}. (e) Find the primary decomposition of R ⁸ under T and the projections on the primary components. Find cyclic decompositions of each primary component. (f) Find the Jordan form of A. Find a basis that puts A in the Jodan form. (g) Find a direct-sum decomposition of R ⁸ into T-cyclic subspaces other than those in the cyclic decomposition, if possible. (h) Show that A is similar to the following matrix B:

.

23. Let T be a triangulable operator on a vector space V over a field F . Prove that given the Jordan form of T it is possible to determine the rational form of T, and conversely. Would you suggest an algorithm for the same?