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linear operator . 線性代數 , 性 .

linear transformation 性 , 再 . 代數

ﬁeld 性 over ﬁeld polynomial ring 代數 (

).

, ,

, . , 性

, . , .

, 大 . 大

, . ,

.

v

(3)

## Form Reduction

linear operator, ordered basis, representative

matrix (form). matrices form similar

matrices. form canonical form ( canonical form

similar), form 數學 .

(大 學 ),

forms.

characteristic polynomial p(x)c linear operator, p(x)

F[x] irreducible polynomial. p(x) forms.

4.1. Diagonal From

, T -invariant subspace , eigenvalue

eigenvector, 再 diagonal form.

linear operator T : V → V, {O} , T -invariant subspace dimension 1 T -invariant subspace. U T -invariant subspace dim(U ) = 1,

v̸= OV U = Span({v}). U T -invariant , T (v)∈ U =

Span({v}). , λ ∈ F T (v) =λv. .

Deﬁnition 4.1.1. T : V → V linear operator, λ ∈ F v∈ V v̸= OV

T (v) =λv, λ T eigenvalue, v T eigenvector.

, eigenvector v v̸= OV, eigenvalue λ λ ̸= 0.

OV T (OV) =λOV, trivial , OV

eigenvector. v̸= OV T (v) = 0v = OV, v Ker(T ) . 0

T eigenvalue, Ker(T )̸= {OV}, T : V → V one-to-one.

65

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Question 4.1. V ﬁnite dimensional vector space T : V → V linear operator.

?

(1) T is an isomorphism (2) T is one-to-one (3) T is onto (4) 0 is not an eigenvalue of T .

linear operator eigenvalue eigenvector, T

eigenvalue, 再 eigenvalue eigenvector . λ

T : V → V eigenvalue, v̸= OV T (v) =λv, λid(v) − T(v) = OV.

v∈ Ker(λid − T), λid − T linear operator isomorphism, Lemma

3.1.4 det(λid − T) = 0. det(λid − T)? , V ordered

basisβ, 再 λid−T β representative matrix [λid − T]β. det(λid−T) det([λid − T]β). dim(V ) = n,

[λid − T]β = [λid]β− [T]β =λ[id]β− [T]β =λIn− [T]β.

λ ∈ F T eigenvalue, det(λIn− [T]β) = 0. T characteristic poly- nomial χT(x) =χ[T ]β(x) = det(xIn− [T]β). , λ ∈ F T eigenvalue, χT(λ) = 0. , λ ∈ F χT(x) = 0 , det(λid − T) = 0. λid − T linear operator one-to-one, v∈ V v̸= OV T (v) =λv.

.

Proposition 4.1.2. V ﬁnite dimensional vector space T : V → V linear operator. λ ∈ F T eigenvalue χT(λ) = 0.

dim(V ) = n , χT(x)∈ F[x]n , F

n ( ), T eigenvalue. λ ∈ F χT(x)

, (x−λ) | χT(x). x−λ F[x] monic irreducible polynomial, χT(x) monic irreducible polynomials χT(x) = p1(x)c1··· pk(x)ck. pi(x)

T eigenvalue. x−λ χT(x)

, .

Deﬁnition 4.1.3. V ﬁnite dimensional F-space, T : V→ V linear operator λ

T eigenvalue. x−λ χT(x) λ algebraic multiplicity.

, χT(x) = p1(x)c1··· pk(x)ck, p1(x), . . . , pk(x) monic irreducible polynomials p1(x) = x−λ, c1 λ algebraic multiplicity.

Question 4.2. T : V → V linear operator dim(V ) = n, T eigenvalue? eigenvalue algebraic multiplicity ?

T eigenvalue , eigenvalue eigenvector

. λ eigenvalue, 前 v̸= OV T (v)−λv = OV v

eigenvalue λ eigenvector. eigenvalue λ eigenvector Ker(T−λid)

OV . vector space.

(5)

Deﬁnition 4.1.4. T : V → V linear operator λ T eigenvalue.

Eλ(T ) = Ker(T−λid) = {v ∈ V | T(v) = λv}.

T λ eigenspace dim(Eλ(T )) λ geometric multiplicity.

v∈ Eλ(T ), T (T (v)) = T (λv) = λT(v), T (v)∈ Eλ(T ). Eλ(T ) T -invariant subspace.

Question 4.3. Lemma 3.5.3 Eλ(T ) T -invariant subspace ? Eλ(T ) ? ordered basisβ [T−λid]β= [T ]β−λIn

matrix, 再 [T ]β−λIn null space N([T ]β−λIn) ={x ∈ Fn| ([T]β−λIn)·x = O}. 再 β

N([T ]β−λIn) V , Eλ(T ) , dim(N([T ]β−λIn)) =

dim(Eλ(T )) λ geometric multiplicity.

Example 4.1.5. T : M2(F)→ M2(F) T

( a b c d

)

=

( a c b d

)

. M2(F) ordered basis β = (

( 1 0 0 0

) ,

( 0 1 0 0

) ,

( 0 0 1 0

) ,

( 0 0 0 1

)

), T β repre-

sentative matrix

[T ]β=



1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1



.

χT(x) =χ[T ]β(x) = (x− 1)3(x + 1). 1 −1 T eigenvalue, algebraic multiplicity 3 1.

T 1 eigenspace E1(T ), N([T ]β− I4),



0 0 0 0

0 −1 1 0

0 1 −1 0

0 0 0 0



 ·



x1 x2 x3 x4



 =



 0 0 0 0



,







0 = 0

−x2+ x3 = 0 x2− x3 = 0

0 = 0

N([T ]β− I4) ={(x1, x2, x2, x4)t| x1, x2, x4∈ F}. 1 geometric multiplicity 3 E1(T ) ={

( x1 x2

x2 x4 )

| x1, x2, x4∈ F}.

, −1 eigenspace E−1(T ), N([T ]β− (−1)I4),



2 0 0 0 0 1 1 0 0 1 1 0 0 0 0 2



 ·



x1 x2 x3 x4



 =



 0 0 0 0



,







2x1 = 0 x2+ x3 = 0 x2+ x3 = 0 2x4 = 0

N([T ]β− (−1)I4) ={(0,x2,−x2, 0)t| x2∈ F}. −1 geometric multiplicity 1 E−1(T ) ={

( 0 x2

−x2 0 )

| x2∈ F}.

Algebraic multiplicity geometric multiplicity, .

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Example 4.1.6. T : P1(F)→ P1(F) T (ax + b) = bx, P1(F) ordered basis β = (x,1). [T ]β =

( 0 1 0 0

)

, χT(x) = x2. 0 T eigenvalue algebraic multiplicity 2. N([T ]β− 0I2) = N([T ]β)

( 0 1 0 0

)

· ( a

b ) ( =

0 0

)

b = 0, N([T ]β− 0I2) ={(a,0)t| a ∈ F}. 0 geometric multplicity 1 E0(T ) ={ax | a ∈ F}.

T eigenvalueλ algebraic multiplicity geometric multiplicity

, . Primary Decomposition Theorem

. Theorem 3.5.8 ,

µT(x) = p1(x)m1··· pk(x)mk and χT(x) = p1(x)c1··· pk(x)ck

p1(x), . . . , pk(x) monic irreducible polynomials λ T eigenvalue, p1(x) = x−λ. Vi= Ker(pi(T )◦mi), for i = 1, . . . , k, Primary Decomposition Theorem (Theorem 3.5.8)

V = V1⊕ ··· ⊕Vk

µT|Vi(x) = pi(x)mi and χT|Vi(x) = pi(x)ci,∀i = 1,...,k.

p1(x) = x−λ,

V1= Ker((T−λid)◦m1)⊇ Ker(T −λid) = Eλ(T ).

λ geometric multiplicity dim(Eλ(T ))≤ dim(V1). , c1 λ algebraic multiplicity, χT|V1(x) = (x−λ)c1, deg(χT|V1(x)) = c1. linear operator characteristic polynomial degree operator space dimension,

dim(V1) = c1. dim(Eλ(T ))≤ c1, .

Lemma 4.1.7. V ﬁnite dimensional F-space, T : V → V linear operator λ T eigenvalue, λ algebraic multiplicity 大 geometric multiplicity.

λ T eigenvalue , Eλ(T ) OV , dim(Eλ(T ))≥ 1,

λ geometric multiplicity 大 1. λ algebraic multiplicity 1, Lemma 4.1.7 λ algebraic multiplicity geometric multiplicity ( λ geometric multiplicity 1). , λ algebraic multiplicity

geometric multiplicity ? .

Proposition 4.1.8. V ﬁnite dimensional F-space, T : V→ V linear operator λ T eigenvalue. λ algebraic multiplicity geometric multiplicity

x−λ | µT(x) (x−λ)2T(x).

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Proof. 前 , µT(x) = p1(x)m1··· pk(x)mk χT(x) = p1(x)c1··· pk(x)ck, p1(x) = x−λ. V1= Ker((T−λid)◦m1). x−λ | µT(x) (x−λ)2T(x),

m1= 1, V1= Ker(T−λid) = Eλ(T ). 前 dim(V1) λ algebraic multiplicity, dim(Eλ(T )) λ geometric multiplicity, .

, λ algebraic multiplicity geometric multiplicity, dim(V1) = dim(Eλ(T )), V1= Ker(T−λid). , v∈ V1, T (v)−λid(v) = OV.

T−λid V1 zero mapping, (T−λid)|V1 = T|V1−λid|V1 = O.

, h(x) = x−λ, h(T|V1) = O. Lemma 3.3.5 T|V1 minimal polynomial µT|V1(x) h(x) = x−λ. Theorem 3.5.8 µT|V1(x) = (x−λ)m1,

m1= 1. 

, χT(x) F[x] , χT(x) =

p1(x)c1··· pk(x)ck, pi(x) x−λi. λi alge-

braic multiplicity geometric multiplicity , Proposition 4.1.8 µT(x) = p1(x)··· pk(x), Vi= Ker(T−λiid) = Eλi(T ), ∀i = 1,...,k. Primary Decomposi- tion Theorem

V = Eλ1(T )⊕ ··· ⊕ Eλk(T ).

V eigenspaces (internal) direct sum. eigenspace

OV T eigenvector, Eλi(T ) basis Si T eigenvector

. V = Eλ1(T )⊕ ··· ⊕ Eλk(T ), Proposition 3.4.6 S1∪ ··· ∪ Sk V

basis, V basis T eigenvector . {v1, . . . , vn} V

basis, vi T eigenvector eigenvalue γi ( γi ),

V ordered basis β = (v1, . . . , vn). i = 1, . . . , n, T (vi) =γivi,

[T ]β=



γ1 O

. ..

O γn



diagonal matrix ( ). .

Deﬁnition 4.1.9. V ﬁnite dimensional F-space T : V → V linear operator.

V basis T eigenvectors , T diagonalizable linear

operator.

linear operator diagonalizable.

Theorem 4.1.10. V ﬁnite dimensional F-space T : V → V linear operator.

.

(1) T diagonalizable linear operator.

(2) V ordered basis β [T ]β diagonal matrix.

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(3) T characteristic polynomial χT(x) F[x] , T eigenvalue algebraic multiplicity geometric multiplicity .

(4) T minimal polynomial µT(x) F[x] monic

.

Proof. 前 (3)⇒ (1) (1)⇒ (2), (2)⇒ (4). dim(V ) = n β V ordered basis

[T ]β=



γ1 O

. ..

O γn



diagonal matrix. λ1, . . . ,λk 1, . . . ,γn} = {λ1, . . . ,λk}. γi

λj γij. χT(x) =χ[T ]β(x) = (x−γ1)···(x −γk) = (x−λ1)c1···(x −λk)ck, ci∈ N. Theorem 3.3.7 ( Theorem 3.3.9) µT(x) = (x−λ1)m1···(x −λk)mk, mi∈ N. h(x) = (x−λ1)···(x −λk), Lemma 3.2.1

h([T ]β) = ([T ]βλ1In)···([T]βλkIn)

=



γ1λ1 O . ..

O γnλ1

···



γ1λk O . ..

O γnλk



=



1λ1)···(γ1λk) O . ..

O (γn−λ1)···(γnλk)



γi λj, j = 1, . . . , k γij, h([T ]β) = O, h(T ) = O.

Lemma 3.3.5 µT(x)| h(x), µT(x) = h(x) = (x−λ1)···(x −λk), T minimal

polynomial µT(x) F[x] monic .

(4)⇒ (3). µT(x) = (x−λ1)···(x −λk), λi∈ F . Theorem 3.3.7, χT(x) = (x−λ1)c1···(x −λk)ck ci∈ N, χT(x) F[x]

. λ1, . . . ,λk T eigenvalues, i = 1, . . . , k (x−λi)|µT(x) (x−λi)2T(x). Proposition 4.1.8 λi algebraic

multiplicity geometric multiplicity . . 

Question 4.4. dim(V ) = n, T : V → V linear operator. T n eigenvalue, T diagonalizable?

Question 4.5. Example 4.1.5 Example 4.1.6 T diagonalizable?

. A∈ Mn(F), eigenvalue eigenvector.

Deﬁnition 4.1.11. A∈ Mn(F). λ ∈ F x∈ Fn x̸= O A· x =λx,

λ A eigenvalue, x T eigenvector.

(9)

A characteristic polynomial χA(x) A eigenvalues λ

N(A−λIn) A λ eigenvector, eigenvalue algebraic multiplicity geometric multiplicity, ... 性 , 再 .

Question 4.6. A∈ Mn(F), λ A eigenvalue, λ algebraic multiplicity geometric multiplicity ? A Lemma 4.1.7 Proposition 4.1.6

?

diagonalizable matrix .

Deﬁnition 4.1.12. A∈ Mn(F). Fn basis A eigenvectors , A diagonalizable matrix.

Theorem 4.1.10 A diagonalizable .

linear operator , 再 .

Theorem 4.1.13. A∈ Mn(F). . (1) A diagonalizable matrix.

(2) P∈ Mn(F) invertible P−1· A · P diagonal matrix.

(3) χA(x) F[x] , A eigenvalue

algebraic multiplicity geometric multiplicity .

(4) µA(x) F[x] monic .

A diagonalizable, Theorem 4.1.13 (2) P−1·A·P diagonal matrix

A diagonal form. P A diagonal form.

P−1· A · P = D =



γ1 O

. ..

O γn

,

Pi∈ Fn P i-th column. 前 i-th column ,

A· P i-th column A· Pi, P· D i-th column γiPi, A· P = P · D A· Pi=λPi, P i-th column Pi eigenvalue γi eigenvector.

diagonalizable matrix A eigenvectors Fn basis,

column column , invertible matrix P, A .

P−1· A · P .

diagonalizable matrices, diagonal form

similar. A diagonalizable, B∼ A, B diagonalizable.

P invertible P−1· A · P = D diagonal matrix. Q invertible B = O−1· A · Q,

(Q−1· P)−1· B · (Q−1· P) = (P−1· Q) · (Q−1· A · Q) · (Q−1· P) = P−1· A · P = D.

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Q−1· P invertible B diagonalizable.

A, B diagonalizable, A∼ B, characteristic poly- nomial, eigenvalues A B eigenvalue algebraic multiplicity

. eigenvalue algebraic multiplicity geometric multiplicity,

A, B diagonal form eigenvalue 線 數 . ,

A, B diagonal form eigenvalue 線 數 ,

diagonal form 線 , diagonal form . 線

eigenvector ordered basis ( (i, i)-th entry ( j, j)-th

entry P i-th column j-th column ), A∼ B.

4.2. Triangular Form

linear operator T characteristic polynomial monic polyno-

mials , T diagonalizable. 探 T

.

χT(x) ( χT(x) =

(x−λ1)c1···(x −λk)ck). V over ﬁeld F algebraically closed ( F =C) . Primary Decomposition Theorem, T minimal polynomial µT(x) = (x−λ)m. (T−λid)◦m= O.

linear operator T : V → V T◦m= O, nilpotent,

m T◦m= O, nilpotent operator index. T−λid nilpotent

index m. 探 nilpotent operator 性 .

linear operator T : V → V. v∈ Im(T◦i), u∈ V v = T◦i(u),

i≥ 2 , v = T◦i−1(T (u))∈ Im(T◦i−1). chain of

subspaces

V ⊇ Im(T) ⊇ Im(T◦2)⊇ ··· ⊇ Im(T◦i−1)⊇ Im(T◦i)⊇ ··· .

, T nilpotent of index m, .

Lemma 4.2.1. dim(V ) > 0, T nilpotent operator of index m, chain of subspaces.

V) Im(T) ) Im(T◦2)) ··· ) Im(T◦i−1)) Im(T◦i)) ··· ) Im(T◦m−1)) Im(T◦m) ={OV}.

Proof. Im(T◦m−1)) Im(T◦m) ={OV}. T◦m= O, v∈ V, T◦m(v) = OV, Im(T◦m) ={OV}. , Im(T◦m−1) = Im(T◦m) ={OV}, T◦m−1= O,

mT◦m= O , Im(T◦m−1)̸= Im(T◦m).

V ) Im(T). Im(T ) = V , v∈ V v1 ∈ V

v = T (v1). v1∈ V, v2∈ V v = T (v1) = T◦2(v2), V = Im(T◦2).

, V = Im(T◦i), ∀i ∈ N. V ̸= {OV}, T nilpotent ,

V ̸= Im(T).

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, 1≤ i ≤ m − 2, v∈ Im(T◦i+1) u∈ V v = T◦i+1(u) = T (T◦i(u)). Im(T◦i) = Im(T◦i+1), T◦i(u)∈ Im(T◦i) = Im(T◦i+1) w∈ V T◦i(u) = T◦i+1(w). v = T (T◦i(u)) = T◦i+2(w)∈ Im(T◦i+2), Im(T◦i+1) = Im(T◦i+2).

Im(T◦m−1) = Im(T◦m), 前 Im(T◦m−1)̸= Im(T◦m) ,

Im(T◦i)̸= Im(T◦i+1), . 

dim(V ) = n T : V → V nilpotent operator of index m, triangular form. Im(T◦m−1) ordered basis (v1, . . . , vk1),

T (vi)∈ Im(T◦m) ={OV},

T (vi) = OV,∀i = 1,...,k1.

{vk1+1, . . . , vk2} (v1, . . . , vk1, . . . , vk2) Im(T◦m−2) ordered basis.

T (vi)∈ Im(T◦m−1) = Span({v1, . . . , vk1}), ∀i = k1+ 1, . . . , k2,

ordered basis (v1, . . . , vk1, . . . , vk2) T|Im(T◦m−2) representative matrix ( Ok1,k1

Ok2−k1,k1 Ok2−k1,k2−k1 )

,

Oi, j i× j , k1× k2− k1 .

{vk2+1, . . . , vk3} (v1, . . . , vk1, . . . , vk2, . . . , vk3) Im(T◦m−3) ordered basis.

T (vi)∈ Im(T◦m−2) = Span({v1, . . . , vk1, . . . , vk2}), ∀i = k2+ 1, . . . , k3,

ordered basis (v1, . . . , vk1, . . . , vk2, . . . , vk3) T|Im(T◦m−3) representative matrix

 Ok1,k1

Ok2−k1,k1 Ok2−k1,k2−k1 Ok3−k2,k1 Ok3−k2,k2−k1 Ok3−k2,k3−k2

.

Im(T ) ordered basis (v1, . . . , vkm−1), j = 1, . . . , m− 1, (v1, . . . , vkj) Im(T◦m− j) ordered basis

T (vi)∈ Im(T◦m−( j−1)) = Span({v1, . . . , vkj−1}), ∀i = kj−1+ 1, . . . , kj. {vkm−1+1, . . . , vn} (v1, . . . , vkm−1, . . . , vn) V ordered basis,

T (vi)∈ Im(T) = Span({v1, . . . , vkm−1}), ∀i = km−1+ 1, . . . , kn, ordered basis (v1, . . . , vkm−1, . . . , vn) T representative matrix



O

... . .. ∗

O O O

.

(12)

Proposition 4.2.2. V ﬁnite dimensional F-space T : V→ V linear operator.

T nilpotent V ordered basis β [T ]β upper triangular matrix

[T ]β0.

Proof. 前 : T nilpotent, V ordered basisβ [T ]β upper triangular matrix 線 0. , [T ]β upper triangular matrix 線 0, χT(x) =χ[T ]β(x) = xn ( n = dim(V )), T◦n= O, T

nilpotent. 

Question 4.7. V ﬁnite dimensional F-space T : V → V nilpotent operator of index m, χT(x) ? µT(x) ?

, linear operator T : V → V, Im(T ), V ordered

basis β, representative matrix [T ]β. 再 [T ]β column space C([T ]β) (

C(A) A column space). column space τβ◦−1 V ,

Im(T ) . upper triangular matrix .

Example 4.2.3. linear operator T : P2(R) → P2(R), T (ax2+ bx + c) = (c−a)x2+ cx + (c− a). P2(R) ordered basisβ = (x2, x, 1), [T ]β=

−1 0 1

0 0 1

−1 0 1

,

χT(x) = x3. [T ]2β =

 0 0 0

−1 0 1

0 0 0

 µT(x) = x3, T nilpotent of index 3. [T ]2β column space Span({(0,1,0)t}), Im(T◦2) = Span({x}). [T ]β column space, Im(T ) = Span({x,x2+ 1}). x2̸∈ Im(T), P2(R) ordered basisβ= (x, x2+ 1, x2).

T (x) = 0, T (x2+ 1) = 1x + 0(x2+ 1) + 0x2, T (x2) = 0x + (−1)(x2+ 1) + 0x2 [T ]β=

 0 1 0 0 0 −1

0 0 0

 diagonal 0 upper triangular matrix.

T minimal polynomial µT(x) = (x−λ)m , T−λ id nilpotent Proposition 4.2.2 ordered basis β [T−λ id]β = U diagonal 0 upper triangular matrix

U =



0 ∗ ∗

... . .. ∗ 0 ··· 0

.

dim(V ) = n, [T−λ id]β= [T ]β−λIn, [T ]β=λIn+U , diagonal λ upper triangular matrix

λIn+U =



λ

. .. ∗

### O

λ

.

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Theorem 4.2.4. V ﬁnite dimensional F-space. T : V→ V linear operator characteristic polynomial

χT(x) = (x−λ1)c1···(x −λk)ck, λ1, . . . ,λk F , V ordered basisβ

[T ]β=

 A1 . ..

## O O

Ak

,

Ai ci× ci upper triangular matrix



λi . .. ∗

## O

λi



.

Proof. Theorem 3.3.9 mi≤ ci µT(x) = (x−λ1)m1···(x−λk)mk, Primary Decomposition Theorem, V = V1⊕··· ⊕Vk, Vi= Ker((T−λiid)◦mi) µT|Vi(x) = (x−λi)mi. T|Viλiid|Vi nilpotent, Proposition 4.2.2, βi Vi

ordered basis, [T|Vi]βi Ai ci×ci upper triangular matrix. β1, . . . ,βk

V ordered basisβ, [T ]β triangular matrix. 

Theorem 4.2.4 T characteristic polynomial F[x]

, T diagonal form triangular form.

T diagonalizable, T◦i. V eigenvectors

ordered basisβ [T ]β =



γ1 O

. ..

O γn

, [T◦i]β =



γ1i O . ..

O γni

.

T diagonal form , trianbular form T◦i.

V T -invariant subspaces direct sum V = V1⊕ ··· ⊕Vk. v∈ V, v = v1+··· + vk, vi∈ Vi (Proposition 3.4.6). i = 1, . . . , k,

linear operator πi : V → V, πi(v) = vi. linear operator the projection to Vi with respect to the direct sum V = V1⊕ ··· ⊕Vk.

v∈ Vi, πi(v) = v. Vi T -invariant, v∈ Vi,

T (v)∈ Vi. v∈ V, v = v1+··· + vk, vi∈ Vi, T (πi(v)) = T (vi), πi(T (v)) =πi(T (v1) +··· + T(vk)) = T (vi).

T◦πii◦ T, ∀i = 1,...,k. (4.1) Theorem 4.2.5. V ﬁnite dimensional F-space. T : V→ V linear operator minimal polynomial

µT(x) = (x−λ1)m1···(x −λk)mk

λ1, . . . ,λk F , T = TD+ TN TD diagonalizable, TN nilpotent of index m = max{m1, . . . , mk}, TD◦ TN= TN◦ TD.

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Proof. Primary Decomposition V = V1⊕ ··· ⊕Vk, Vi= Ker((T−λiid)◦mi), πi: V→ V the projection to Vi with respect to the direct sum V = V1⊕ ··· ⊕Vk. V

linear operator TD1π1+··· +λkπk. vi∈ Vi, TD(vi) =λivi,

Vi basis, TD eigenvectors . V V1, . . . ,Vk direct sum, Vi

basis V basis. V basis TD eigenvectors , TD

diagonalizable.

TN = T− TD V linear operator. vi∈ Vi, TN(vi) = T (vi)− TD(vi) = T (vi)λivi ∈ Vi, Vi TN-invariant. µT|Vi(x) = (x−λi)mi, mi

(T−λiid)◦mi(vi) = OV, ∀vi ∈ Vi, µTN|Vi(x) = xmi. Lemma 3.5.6 µTN(x) = lcm(xm1, . . . , xmk) = xm, m = max{m1, . . . , mk}, TN nilpotent of index m.

TD◦ T = (λ1π1+··· +λkπk)◦ T =λ11◦ T) + ··· +λkk◦ T), (4.1)

T◦ TD1(T◦π1) +··· +λk(T◦πk) = TD◦ T.

TD◦ TN= TD◦ (T − TD) = TD◦ T − TD◦ TD= T◦ TD− TD◦ TD= (T− TD)◦ TD= TN◦ TD.

 Question 4.8. Theorem 4.2.4 ordered basis β, [T ]β upper triangular matrix, Theorem 4.2.5 TD, TN β representative matrix [TD]β, [TN]β ? Question 4.9. Theorem 4.2.5, T minimal polynomial µT(x)

F[x] monic , T diagonalizable?

Theorem 4.2.5, triangular form T◦i . TD◦TN= TN◦TD, T◦2= (TD+ TN)◦ (TD+ TN) = TD◦2+ TD◦ TN+ TN◦ TD+ TN◦2= TD◦2+ 2TD◦ TN+ TN◦2.

T◦i=

### ∑

i j=0

(i j

)

TD◦i− j◦ TN◦ j.

TD diagonalizable TD◦ j, TN nilpotent of index m,

j≥ m, TN◦ j= O. T◦i .

linear operator n× n matrix .

Theorem 4.2.6. A∈ Mn(F) characteristic polynomial minimal polynomial

χA(x) = (x−λ1)c1···(x −λk)ck,µA(x) = (x−λ1)m1···(x −λk)mk

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λ1, . . . ,λk F . invertible matrix P

P−1· A · P =

 A1 . ..

## O O

Ak

,

Ai ci× ci upper triangular matrix



λi . .. ∗

### O

λi

.

P−1· A · P = D + N, D diagonal matrix, N nilpotent matrix D· N = N · D Nm= O, m = max{m1, . . . , mk}.

A triangular form P−1· A · P = D + N, D· N = N · D, P−1· Ai· P =

### ∑

i

j=0

(i j

)

Di− j· Nj.

A∈ Mn(F) χA(X ) = (x−λ1)c1···(x −λk)ck, µA(x) = (x−λ1)m1···(x −λk)mk. invertible matrix M M−1· A · M upper triangular matrix.

Chapter 3 primary decomposition invertible matrix P P−1· A · P block diagonal matrix



A1

. ..

### O

Ak



,

ci× ci matrix Ai. µAi(x) = (x−λi)mi, AiλiIci nilpotent of index mi, Proposition 4.2.2 (AiλiIci)mi−1 column space

basis ( Proposition 4.2.2 Im(T◦m−1) basis), 大 (AiλiIci)mi−2

column space basis, 大 Fci basis. basis

column by column ci× ci matrix Qi, Q−1i · Ai· Qi upper

triangular matrix. Qi diagonal , n× n invertible

matrix 



Q1

. ..

### O

Qk



,

(P· Q)−1· A · (P · Q) = Q−1· (P−1· A · P) · Q =



Q−11 · A1· Q1

. ..

### O

Q−1k · Ak· Qk



,

upper triangular matrix . .

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Example 4.2.7. Example 3.5.10 5× 5 matrix

A =





2 1 1 1 0

1 4 2 2 1

−1 −2 0 −1 −1

0 0 0 1 1

0 −1 −1 −1 0





χA(x) =µA(x) = (x− 1)3(x− 2)2 Q[x] , invertible matrix M∈ M5(Q) M−1· A · M upper triangular matrix.

Example 3.5.10 P∈ M5(Q) A block diagonal matrix.

P−1· A · P =





0 −1 −1 0 0

1 1 0 0 0

0 1 2 0 0

0 0 0 1 −1

0 0 0 1 3





.

B =

 0 −1 −1

1 1 0

0 1 2

, C =(

1 −1

1 3

)

triangular forms. µB(x) = (x− 1)3, B− I3 nilpotent matrix.

B− I3=

−1 −1 −1

1 0 0

0 1 1

, (B −I3)2=

 0 0 0

−1 −1 −1

1 1 1

.

Proposition 4.2.2 (B−I3)2 column space basis, w1= (0,−1,1)t, 再 B−I3 column space w2 {w1, w2} B−I3 column space basis,

w2= (−1,1,0)t. 再 w3∈ Q3 {w1, w2, w3} Q3 basis, w3= (0, 0, 1)t. Bw1 = w1, Bw2= w1+ w2, Bw3= w1+ w2+ w3, Q1=

 0 −1 0

−1 1 0

1 0 1

, Q−11 · B · Q1=

 1 1 1 0 1 1 0 0 1

 upper triangular matrix.

µC(x) = (x− 2)2, C− 2I2 nilpotent matrix. C− 2I2= ( −1 −1

1 1

)

, u1=

( −1 1

)

C− 2I2 basis, 再 u2= ( 1

0 )

{u1, u2} Q2 basis. Cu1= 2u1, Cu2= u1+ 2u2, Q2=

( −1 1 1 0

)

, Q−12 ·C · Q2= ( 2 1

0 2 )

upper triangular matrix. Q1, Q2 5× 5 invertible matrix

Q =





0 −1 0 0 0

−1 1 0 0 0

1 0 1 0 0

0 0 0 −1 1

0 0 0 1 0





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upper triagular matrix

(P· Q)−1· A · (P · Q) = Q−1· (P−1· A · P) · Q =





1 1 1 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 2 1 0 0 0 0 2





.

4.3. Jordan Form

Triangular form similar.

ordered basis Jordan form. χT(x)

( χT(x) = (x−λ1)c1···(x −λk)ck). nilpotent .

linear operator T : V → V.T, T◦2, . . . kernel . v∈ Ker(T◦i), T◦i(v) = OV, T◦i+1(v) = T (T◦i(v)) = OV.

chain of subspaces

{OV} ⊆ Ker(T) ⊆ Ker(T◦2)⊆ ··· ⊆ Ker(T◦i−1)⊆ Ker(T◦i)⊆ ··· .

, T nilpotent of index m, Ker(T◦i+1)̸= Ker(T◦i), ∀i = 1,...,m − 1.

Lemma 4.3.1. dim(V ) > 0, T nilpotent operator of index m, chain of subspaces.

{OV} ( Ker(T) ( Ker(T◦2)( ··· ( Ker(T◦i−1)( Ker(T◦i)( ··· ( Ker(T◦m−1)( Ker(T◦m) = V.

Proof. Ker(T◦m−1)( Ker(T◦m) = V . T◦m= O, v∈ V, T◦m(v) = OV, Ker(T◦m) = V . , Ker(T◦m−1) = Ker(T◦m) = V , T◦m−1= O,

mT◦m= O , Ker(T◦m−1)̸= Ker(T◦m).

{OV} ( Ker(T). v∈ V OV = T◦m(v) = T (T◦m−1(v)) T◦m−1(v)∈ Ker(T). Ker(T ) ={OV}, v∈ V T◦m−1(v) = OV T◦m−1= O , Ker(T )̸= {OV}.

, 1≤ i ≤ m − 2, v∈ V OV = T◦m(v) = T◦i+1(T◦m−(i+1)(v)), T◦m−(i+1)(v)∈ Ker(T◦i+1). Ker(T◦i) = Ker(T◦i+1), T◦m−(i+1)(v)∈ Ker(T◦i)

OV = T◦i(T◦m−(i+1)(v)) = T◦m−1(v). v∈ V , T◦m−1= O

, Ker(T◦i)̸= Ker(T◦i+1). 

i≥ 2, v1, . . . , vs∈ Ker(T◦i+1) linearly independent Span(v1, . . . , vs)∩ Ker(T◦i) ={OV}, T (v1), . . . , T (vs)∈ Ker(T◦i) linearly independent.

r1T (v1) +··· + rsT (vs) = OV,

at each point of estimation, form a linear combination of a preliminary esti- mator evaluated at nearby points with the coefficients specified so that the asymptotic bias

Theorem 1.5.9 ﬁnite dimensional vector space linearly independent set 大 basis; spanning

vector space subspaces internal direct sum, 再 external direct. sum, 再 internal

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