linear operator

(1)

大學數學

(2)

大學線性代數 .

linear operator . 線性代數 , 性 .

linear transformation 性 , 再 . 代數

ﬁeld 性 over ﬁeld polynomial ring 代數 (

).

, ,

代. , .

, . , 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.

前 Primary Decomposition Theorem, linear operator

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.

O_V T (O_V) =λOV, trivial , O_V

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

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

65

(4)

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 polynomial 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] 數 n , F 數

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)^c^k. p_i(x) 數

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)^c^k, p₁(x), . . . , p_k(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)

O_V . vector space.

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

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

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

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

Question 4.3. Lemma 3.5.3 E_λ(T ) T -invariant subspace ? E_λ(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 , E_λ(T ) , dim(N([T ]_β−λIn)) =

dim(E_λ(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)^m^k and χT(x) = p1(x)^c¹··· pk(x)^c^k

p1(x), . . . , pk(x) monic irreducible polynomials λ T eigenvalue, p₁(x) = x−λ. V_i= Ker(p_i(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(E_λ(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(E_λ(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 , E_λ(T ) O_V , dim(E_λ(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)²-µT(x).

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Proof. 前 , µT(x) = p₁(x)^m¹··· pk(x)^m^k χT(x) = p₁(x)^c¹··· pk(x)^c^k, p1(x) = x−λ. V1= Ker((T−λid)^◦m¹). x−λ | µT(x) (x−λ)²-µT(x),

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

, λ algebraic multiplicity geometric multiplicity, dim(V₁) = dim(E_λ(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)^c^k, p_i(x) x−λi. λi alge-

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

V = E_λ₁(T )⊕ ··· ⊕ E_λ_k(T ).

V eigenspaces (internal) direct sum. eigenspace

O_V T eigenvector, E_λ_i(T ) basis Si T eigenvector

. V = E_λ₁(T )⊕ ··· ⊕ E_λk(T ), Proposition 3.4.6 S1∪ ··· ∪ Sk V

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

basis, v_i T eigenvector eigenvalue γi ( γi ),

V ordered basis β = (v1, . . . , v_n). i = 1, . . . , n, T (v_i) =γiv_i,

[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 {γ1, . . . ,γn} = {λ1, . . . ,λk}. γi

λj γi=λj. χT(x) =χ[T ]_β(x) = (x−γ1)···(x −γk) = (x−λ1)^c¹···(x −λk)^c^k, ci∈ N. Theorem 3.3.7 ( Theorem 3.3.9) µT(x) = (x−λ1)^m¹···(x −λk)^m^k, m_i∈ N. h(x) = (x−λ1)···(x −λk), Lemma 3.2.1

h([T ]_β) = ([T ]_β−λ1I_n)···([T]_β−λkI_n)

=





γ1−λ1 O . ..

O γn−λ1



···





γ1−λk O . ..

O γn−λk





=





(γ1−λ1)···(γ1−λk) O . ..

O (γn−λ1)···(γn−λk)





γi λj, j = 1, . . . , k γi=λj, 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)^c^k ci∈ N, χT(x) F[x]

. λ1, . . . ,λk T eigenvalues, i = 1, . . . , k (x−λi)|µT(x) (x−λi)²-µT(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?

前 linear operator, 性 n× n

. 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.

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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



,

P_i∈ 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 polynomial, 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)^c^k). 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, 數

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) = O_V, Im(T^◦m) ={OV}. , Im(T^◦m−1) = Im(T^◦m) ={OV}, T^◦m−1= O,

m 數 T^◦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, . . . , v_k₁),

T (vi)∈ Im(T^◦m) ={OV},

T (vi) = OV,∀i = 1,...,k1.

{vk1+1, . . . , v_k₂} (v₁, . . . , v_k₁, . . . , v_k₂) Im(T^◦m−2) ordered basis.

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

ordered basis (v₁, . . . , v_k₁, . . . , v_k₂) T|Im(T^◦m−2) representative matrix ( O_k₁_,k₁ ∗

O_k₂_−k₁_,k₁ O_k₂_−k₁_,k₂_−k₁ )

,

O_{i, j} i× j , ∗ k1× k2− k1 .

{vk2+1, . . . , v_k₃} (v₁, . . . , v_k₁, . . . , v_k₂, . . . , v_k₃) Im(T^◦m−3) ordered basis.

T (vi)∈ Im(T^◦m−2) = Span({v1, . . . , v_k₁, . . . , v_k₂}), ∀i = k2+ 1, . . . , k₃,

ordered basis (v1, . . . , v_k₁, . . . , v_k₂, . . . , v_k₃) T|Im(T^◦m−3) representative matrix



 O_k₁_,k₁ ∗ ∗

O_k₂_−k₁_,k₁ O_k₂_−k₁_,k₂_−k₁ ∗ O_k₃_−k₂_,k₁ O_k₃_−k₂_,k₂_−k₁ O_k₃_−k₂_,k₃_−k₂



.

Im(T ) ordered basis (v1, . . . , vkm−1), j = 1, . . . , m− 1, (v₁, . . . , v_k_j) Im(T^{◦m− j}) ordered basis

T (v_i)∈ Im(T^{◦m−( j−1)}) = Span({v1, . . . , v_k_j−1}), ∀i = kj−1+ 1, . . . , k_j. {vk_m−1+1, . . . , v_n} (v₁, . . . , v_k_m−1, . . . , v_n) V ordered basis,

T (v_i)∈ Im(T) = Span({v1, . . . , v_k_m−1}), ∀i = km−1+ 1, . . . , k_n, ordered basis (v₁, . . . , v_k_m−1, . . . , v_n) T representative matrix





O ∗ ∗

... . .. ∗

O O O



.

線 0 upper triangular matrix ( ), .

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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 ]_β 線 0.

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 τ_β^◦−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 ]²_β =



 0 0 0

−1 0 1

0 0 0



 µT(x) = x³, T nilpotent of index 3. [T ]²_β 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



.

(13)

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

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

[T ]_β=



 A₁ _{. ..}

O O

^A^k



,

A_i c_i× ci upper triangular matrix







λi ∗ ∗ . .. ∗

O

^λⁱ





.

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

ordered basis, [T|Vi]_β_i A_i c_i×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, v_i∈ Vi (Proposition 3.4.6). i = 1, . . . , k,

linear operator πi : V → V, πi(v) = vi. linear operator the projection to V_i 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, v_i∈ Vi, T (πi(v)) = T (vi), πi(T (v)) =πi(T (v₁) +··· + T(vk)) = T (v_i).

T◦πi=πi◦ 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)^m^k

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

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

linear operator TD=λ1π1+··· +λkπk. v_i∈ Vi, TD(vi) =λiv_i,

V_i basis, T_D eigenvectors . V V₁, . . . ,V_k direct sum, V_i

basis V basis. V basis TD eigenvectors , TD

diagonalizable.

T_N = T− TD V linear operator. v_i∈ Vi, T_N(v_i) = T (v_i)− TD(v_i) = T (v_i)−λiv_i ∈ Vi, Vi TN-invariant. µT|_Vi(x) = (x−λi)^mⁱ, mi

數 (T−λiid)^◦mⁱ(vi) = OV, ∀vi ∈ Vi, µTN|_Vi(x) = x^mⁱ. Lemma 3.5.6 µTN(x) = lcm(x^m¹, . . . , x^m^k) = x^m, m = max{m1, . . . , m_k}, T_N nilpotent of index m.

T_D◦ T = (λ1π1+··· +λkπk)◦ T =λ1(π1◦ T) + ··· +λk(πk◦ T), (4.1)

T◦ TD=λ1(T◦π1) +··· +λk(T◦πk) = TD◦ T.

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

Question 4.8. Theorem 4.2.4 ordered basis β, [T ]_β upper triangular matrix, Theorem 4.2.5 T_D, T_N β representative matrix [T_D]_β, [T_N]_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= T_N◦TD, T^◦2= (TD+ TN)◦ (TD+ TN) = TD◦2+ TD◦ TN+ TN◦ TD+ TN◦2= TD◦2+ 2TD◦ TN+ TN◦2.

數學

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)^c^k,µA(x) = (x−λ1)^m¹···(x −λk)^m^k

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λ1, . . . ,λk F . invertible matrix P

P⁻¹· A · P =



 A₁ _{. ..}

O O

^A^k



,

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 N^m= O, m = max{m1, . . . , m_k}.

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

∑

ⁱ

j=0

(i j

)

Dⁱ^{− j}· N^j.

A∈ Mn(F) χA(X ) = (x−λ1)^c¹···(x −λk)^c^k, µA(x) = (x−λ1)^m¹···(x −λk)^m^k. 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

. ..

O

^Ak





,

ci× ci matrix Ai. µAi(x) = (x−λi)^mⁱ, Ai−λiIci nilpotent of index m_i, Proposition 4.2.2 (A_i−λiI_c_i)^mⁱ⁻¹ column space

basis ( Proposition 4.2.2 Im(T^◦m−1) basis), 大 (A_i−λiIci)^mⁱ⁻²

column space basis, 大 F^cⁱ basis. basis

column by column c_i× ci matrix Q_i, Q⁻¹_i · Ai· Qi upper

triangular matrix. Qi diagonal , n× n invertible

matrix 





Q1

O

. ..

O

^Qk





,

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







Q⁻¹₁ · A1· Q1

O

. ..

O

^Q⁻¹_k · 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)^c^k). nilpotent .

linear operator T : V → V. 探 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) = O_V, Ker(T^◦m) = V . , Ker(T^◦m−1) = Ker(T^◦m) = V , T^◦m−1= O,

m 數 T^◦m= O , Ker(T^◦m−1)̸= Ker(T^◦m).

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

, 1≤ i ≤ m − 2, v∈ V O_V = 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)

O_V = 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, . . . , v_s)∩ Ker(T^◦i) ={OV}, T (v1), . . . , T (vs)∈ Ker(T^◦i) linearly independent.

r₁T (v₁) +··· + rsT (v_s) = O_V,