Linear Operator

大學線性代數再探

大學數學

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

Chapter 3

Linear Operator

V vector space , V V linear transformation, linear

operator on V . T : V → V linear operator ,

T^◦2= T◦ T, T^◦3= T◦ T^◦2, . . . i∈ N T^◦i= T◦ T^◦i−1

(T^◦0= id). V 代數 ( F[T ]-module),

T V . linear operator . 大

代數 , 代數 言, 大 (

, ) .

3.1. Basic Concept

linear operator linear transformation 前 .

vector space, ordered basis,

. T : V→ V linear operator, T representative

matrix, V ordered basisβ = (v1, . . . , v_n), β, β[T ]_β

n× n matrix. ordered basis , T representative matrix,

[T ]_β ,

[T ]_β =(

τβ(T (v₁)), . . . ,τβ(T (v_n))) .

T1, T₂ V linear operator, Chapter 2 Proposition 2.4.5

[T2◦ T1]_β= [T2]_β· [T1]_β. (3.1) , [id]_β = I_n.

V linear operators vector space L (V,V) L (V).

n×n , M_n(F) over F n×n matrices.

, V ordered basisβ , Theorem 2.4.4

L (V) M_n(F) isomorphism, Φ : L (V) → Mn(F), T 7→ [T]_β. [id]_β = I_n, [T ]_β= I_n T = id. [T ]_β zero matrix T : V→ V zero 41

mapping, T (v) = O_V,∀v ∈ V. , zero matrix zero mapping

O . .

Lemma 3.1.1. V finite dimensional vector space, dim(V ) = n β V ordered basis. T : V → V linear operator, :

[T ]_β= In⇔ T = id and [T]_β = O⇔ T = O.

前 linear operator , ordered

basis. , identity linear operator, id : V→ V. linear

operator ordered basis , representative matrix

, matrices , _β′[id]_β change of basis matrix

. Proposition 2.4.6, .

Lemma 3.1.2. β,β^′ V ordered bases, T : V → V linear operator, [T ]_β′=_β[id]⁻¹_β′ · [T]_β·_β[id]_β′.

Proof. Proposition 2.4.6, [T ]_β′ =_β′ [id]_β· [T]_β·_β[id]_β′. dim(V ) = n, (2.6)

β^′[id]_β·_β[id]_β′=_β[id]_β′·_β^′[id]_β = In,

β^′[id]_β=_β[id]⁻¹_β′ , . 

A, B∈ Mn(F), P M_n(F) invertible matrix, B = P⁻¹· A · P, A, B similar matrix, A∼ B . det(P⁻¹) = det(P)⁻¹

det(B) = det(P⁻¹· A · P) = det(P⁻¹) det(A) det(P) = det(A).

Lemma 3.1.2 [T ]_β ∼ [T]_β^′, det([T ]_β) = det([T ]_β′).

ordered basis, T representative matrix determinant , T determinant, det(T ) = det([T ]_β).

Lemma 3.1.2 ? A∼ [T]_β, V ordered basis

β^′ A = [T ]_β′ ? , P invertible matrix A = P⁻¹· [T]_β· P, Proposition 2.4.7, V ordered basis β^′ P =_β [id]_β′, Lemma 3.1.2

A = P⁻¹· [T]_β· P =_β [id]⁻¹_β′ · [T]_β·_β[id]_β′ = [T ]_β′. .

Proposition 3.1.3. V finite dimensional vector space, dim(V ) = n β V ordered basis. T : V → V linear operator A∈ Mn(F), A∼ [T]_β

V ordered basisβ^′ A = [T ]_β′.

3.1. Basic Concept 43

探 linear operator 性 , ordered basis

square matrix , Proposition 3.1.3 性 similar matrices

, . .

Lemma 3.1.4. V finite dimensional vector space, β V ordered basis

T : V → V linear operator, :

(1) T isomorphism.

(2) [T ]_β invertible matrix.

(3) det(T )̸= 0.

Proof. [T ]_β invertible matrix det([T ]_β)̸= 0, (1)⇔ (2).

dim(V ) = n, T isomorphism, T^◦−1 linear operator, [T^◦−1]_β· [T]_β = [id]_β= [T ]_β· [T^◦−1]_β [id]_β = I_n

[T ]_β invertible. , A· [T]_β = I_n, Φ : L (V) → Mn(F), isomorphism, T^′: V → V Φ(T^′) = [T^′]_β= A. [T^′◦ T]_β = [T^′]_β· [T]_β = I_n Lemma 3.1.1 T^′◦ T = id, [T ]_β· [T^′]_β= I_n T◦ T^′= id, T isomorphism.  Question 3.1. Lemma 3.1.4 A∼ B A invertible B invertible.

linear operator 性 determinant,

determinant . A∈ Mn(F), a_ik∈ F A (i, k)-th entry (

A i-th row k-th column ), A_ik∈ Mn−1(F) A i-th row

k-th column (n− 1) × (n − 1) matrix. det(A)

i-th row ,

det(A) =

∑

(−1)^i+kaikdet(Aik), j-th column

det(A) =

∑

(−1)^{k+ j}a_{k j}det(A_{k j}).

n×n matrix adjoint matrix of A, adj(A) , adj(A) (i, j)-th entry

adj(A)i j= (−1)^{i+ j}det(Aji).

.

Lemma 3.1.5. A n× n matrix, adj(A) A adjoint matrix, A· adj(A) = adj(A) · A = det(A)In.

Proof. det(A)I_n diagonal matrix, 線 det(A) 線 0. A· adj(A) (i, i)-th entry,

∑

a_ikadj(A)_ki=

∑

(−1)^k+ia_ikdet(A_ik) = det(A).

i̸= j, A · adj(A) (i, j)-th entry,

∑

aikadj(A)k j=

∑

(−1)^{k+ j}aikdet(Ajk).

A j-th row i-th row 代, A^′ , A^′ i-th row j-th

row , det(A^′) = 0. A^′ ( j, k)-th entry a^′_jk a_ik A^′_jk= A_jk, A^′ j-th row ,

0 = det(A^′) =

∑

(−1)^j+ka^′_jkdet(A^′_jk) =

∑

(−1)^j+ka_ikdet(A_jk), i̸= j A· adj(A) (i, j)-th entry

∑

aikadj(A)k j=

∑

(−1)^{k+ j}aikdet(Ajk) = 0.

A· adj(A) = det(A)In. , column determinant, adj(A)· A =

det(A)In. 

3.2. Characteristic Polynomial

前 linear operator , square matrix ,

探 n× n matrix, 再 linear operator .

數 F polynomial f (x) = cdx^d+··· + c1x + c0 n× n matrix A,

f (A) = c_dA^d+··· + c1A + c₀I_n.

, f (A) n× n matrix. , Aⁱ f (A)

.

Aⁱ· f (A) = Aⁱ· (cdA^d+··· + c1A + c0In)

= c_dA^d+i+··· + c1A¹⁺ⁱ+ c₀Aⁱ= (c_dA^d+··· + c1A + c₀I_n)· Aⁱ= f (A)· Aⁱ.

, .

Lemma 3.2.1. f (x), g(x), h(x)∈ F[x] f (x) = g(x)h(x). A∈ Mn(F), g(A)· h(A) = h(A) · g(A) = f (A).

再 A , g(x), h(x)∈ F[x]

A, B∈ Mn(F), g(A)· h(B) = h(B) · g(A).

A∼ B, f A)∼ f (B) ? P invertible,

(P⁻¹· A · P)²= (P⁻¹· A · P) · (P⁻¹· A · P) = P⁻¹· A²· P.

3.2. Characteristic Polynomial 45

數學

(P⁻¹· A · P)ⁱ= P⁻¹· Aⁱ· P.

.

Lemma 3.2.2. f (x)∈ F[x] A, B∈ Mn(F). A∼ B, f (A)∼ f (B).

Proof. A∼ B P invertible B = P⁻¹· A · P. f (x) = cdx^d+··· + c1x + c0,

f (B) = cdB^d+··· + c1B + c0In= cd(P⁻¹· A · P)^d+··· + c1(P⁻¹· A · P) + c0In

= c_d(P⁻¹·A^d·P)+···+c1(P⁻¹·A·P)+c0I_n= P⁻¹·(cdA^d+···+c1A + c₀I_n)·P = P⁻¹· f (A)·P,

f (A)∼ f (B). 

linear operator, f (x) = c_dx^d+··· + c1x + c₀∈ F[x]

T : V → V linear operator, linear operators

( (3.1)),

f (T ) = cdT^◦d+··· + c1T + c0id,

f (T ) V V linear operator. T^◦i◦ f (T) = f (T) ◦ T^◦i, .

Lemma 3.2.3. f (x), g(x), h(x)∈ F[x] f (x) = g(x)· h(x). T ∈ L (V), g(T )◦ h(T) = h(T) ◦ g(T) = f (T).

f (x) = g(x)· h(x) f (T ) = g(T )◦ h(T) g(h(T )).

g(T ) h(T ) linear operator f (T ) operator, h(T )

linear operator 代 g(x) .

V ordered basis β F(T ) representative matrix T

representative matrix . 再 3.1, [T^◦2]_β= [T ]²_β, 數學

[T^◦i]_β= [T◦ T^◦i−1]_β= [T ]_β· [T]ⁱ_β⁻¹= [T ]ⁱ_β, .

Lemma 3.2.4. V finite dimensional F-space, β V ordered basis T : V → V linear operator. f (x) = c_dx^d+··· + c1x + c0∈ F[x],

[ f (T )]_β = f ([T ]_β) = c_d[T ]^d_β+··· + c1[T ]_β+ c₀I_n.

Proof. [ f (T )]_β f (T ) representative matrix, Φ linear transformation,

[ f (T )]_β= [c_dT^◦d+··· + c1T + c₀id]_β =

cd[T^◦d]_β+··· + c1[T ]_β+ c₀[id]_β= c_d[T ]^d_β+··· + c1[T ]_β+ c₀In= f ([T ]_β).

 n× n matrix . dim(Mn(F)) = n², A∈ Mn(F), S = {In, A, A², . . . , Aⁿ²}. #(S) = n²+ 1 > dim(M_n(F)), S linearly dependent.

c0, c1, . . . , c_n² ∈ F 0

c_n²Aⁿ²+··· + c1A + c0In= O.

f (x) = c_n2xⁿ²+··· + c1x + c₀, f (A) = O. : n× n matrix

A, 數 大 n² f (x)∈ F[x] f (A) n× n zero matrix

O. c_n² 0 deg( f (x)) = n², c_n², . . . , c1, c0 0,

f (x) .

Question 3.2. A∼ B f (x)∈ F[x] f (A) = O, f (B) = O?

Question 3.3. dim(V ) = n T : V→ V linear operator, nonzero polynomial f (x)∈ F[x] deg( f (x))≤ n² f (T ) = O?

數 n f (x) f (A) = O, characteristic

polynomial.

Definition 3.2.5. A∈ Mn(F), χA(x) = det(xI_n−A) ∈ F[x], A characteristic polynomial.

det(A−xIn) A characteristic polynomial, det(xIn−A)

χA(x) monic polynomial ( 數 1). determinant

數學 , A n×n matrix ,χA(x) 數 n 數 1.

χA(x) ( xⁿ⁻¹ ) 數 −tr(A) ( : tr(A) A trace, 線 ). x = 0 代 χA(x) χA(x) 數 χA(0) = det(−A) = (−1)ⁿdet(A).

Example 3.2.6. xIn− In= (x− 1)In, χIn(x) = det((x− 1)In) = (x− 1)ⁿ. 2× 2 matrix characteristic polynomial.

A₁=

( 1 −1 1 −1

) , A₂=

( 1 −1 0 −1

) , A₃=

( 1 −1 2 −1

) ,

χA1 = det

( x− 1 1

−1 x + 1 )

= (x− 1)(x + 1) + 1 = x², χA2 = det

( x− 1 1 0 x + 1

)

= (x− 1)(x + 1) = x²− 1, χA3 = det

( x− 1 1

−2 x + 1 )

= (x− 1)(x + 1) + 2 = x²+ 1.

Question 3.4. χI2(I₂), χA1(A₁), χA2(A₂), χA3(A₃) .

similar matrices characteristic polynomial .

3.2. Characteristic Polynomial 47

Proposition 3.2.7. A, B∈ Mn(F) A∼ B, χA(x) =χB(x).

Proof. A∼ B invertible matrix P B = P⁻¹· A · P. xIn diagonal matrix, xI_n· P = P · xIn, P⁻¹· xIn· P = xIn.

xIn− B = xIn− P⁻¹· A · P = P⁻¹· xIn· P − P⁻¹· A · P = P⁻¹· (xIn− A) · P.

χB(x) = det(xIn− B) = det(P⁻¹· (xIn− A) · P) = det(P)⁻¹det(xIn− A)det(P) =χA(x).

 , T : V→ V linear operator, β,β^′ V ordered bases, [T ]_β ∼ [T ]_β′, Proposition 3.2.7 χ[T ]_β(x) =χ[T ]_β′(x). linear operator characteristic polynomial.

Definition 3.2.8. V finite dimensional F-space. V linear operator T : V→ V , V ordered basis β, T characteristic polynomial χ[T ]_β(x), χT(x) .

A characteristic polynomial xI_n− A ,

entry , 探 . ,

x^dAd+··· + xA1+ A0, Ai∈ Mn(F) . ( 5x²+ 3 4x− 1

7 x³− 2x²+ x )

= x³

( 0 0 0 1

) + x²

( 5 0 0 −2

) + x

( 0 4 0 1

) +

( 3 −1 7 0

) .

F 代 x, xA 數 x A.

A, B∈ Mn(F), (rA)· (sB) = (rs)A · B,

(xⁱA)· (x^jB) = x^{i+ j}A· B.

,

(A + xB)²= (A + xB)· (A + xB) = A²+ A· (xB) + xB · A + (xB)²= A²+ x(A· B + B · A) + x²B², , (A + xB)² A²+ 2x(A· B) + x²B².

entry square matrices ,

. x , .

, . .

Example 3.2.9.

( 5x²+ 3 4x− 1

7 x

)

= x²

( 5 0 0 0

) + x

( 0 4 0 1

) +

( 3 −1 7 0

) . ( x− 1 1

−x x + 2 )

= x

( 1 0

−1 1 )

+

( −1 1

0 2

) .

( 5x²+ 3 4x− 1

7 x

)

·

( x− 1 1

−x x + 2 )

=

( 5x³− 9x²+ 4x− 3 9x²+ 7x + 1

−x²+ 7x− 7 x²+ 2x + 7 )

,

( x²

( 5 0 0 0

) + x

( 0 4 0 1

) +

( 3 −1

7 0

))

· (

x

( 1 0

−1 1 )

+

( −1 1

0 2

))

= x³

( 5 0 0 0

)( 1 0

−1 1 )

+ x²

(( 5 0 0 0

)( −1 1

0 2

) +

( 0 4 0 1

)( 1 0

−1 1 ))

+ x

(( 0 4 0 1

)( −1 1

0 2

) +

( 3 −1

7 0

)( 1 0

−1 1 ))

+

( 3 −1

7 0

)( −1 1 0 2

)

= x³

( 5 0 0 0

) + x²

( −9 9

−1 1 )

+ x

( 4 7 7 2

) +

( −3 1

−7 7 )

.

x^dAd+··· + xA1+ A0= x^dBd+··· + xB1+ B0, Ai, Bi∈ Mn(F),

A_i= B_i,∀i = 0,1,...,d. A_i̸= Bi, entry xⁱ

數 , . , characteristic polynomial

性 .

Theorem 3.2.10 (Cayley-Hamilton Theorem). A∈ Mn(F), χA(x) A characteristic polynomial, χA(A) = O.

Proof. χA(x) = xⁿ+··· + c1x + c₀. xI_n− A adjoint matrix, Lemma 3.1.5

adj(xI_n− A) · (xIn− A) = det(xIn− A)In=χA(x)I_n= xⁿI_n+··· + xc1I_n+ c₀I_n.

xIn−A i-th row k-th column , (n−1)×(n−1) matrix determinant

數 n , adjoint matrix adj(A− xIn) entry 數

n , adj(A− xIn) = xⁿ⁻¹B_n−1+··· + xB1+ B₀, B_i∈ Mn(F).

(xⁿ⁻¹Bn−1+ xⁿ⁻²Bn−2+··· + xB1+ B0)· (xIn− A) = xⁿIn+ xⁿ⁻¹cn−1In+··· + xc1In+ c0In (3.2)

(3.2) ,

(xⁿ⁻¹Bn−1+ xⁿ⁻²Bn−2+··· + xB1+ B₀)· (xIn− A)

= xⁿ(Bn−1· In) + xⁿ⁻¹(Bn−2· In− Bn−1· A+) + ··· + x(B0· In− B1· A+) − B0· A

3.3. Minimal Polynomial 49

(3.2) , 數

−B0· A = c0I_n B0· In− B1· A = c1In

...

B_n−2· In− Bn−1· A = cn−1I_n Bn−1· In = In

, A, A², . . . , Aⁿ,

−B0· A = c0I_n B0· A − B1· A² = c1A

...

Bn−2· Aⁿ⁻¹− Bn−1· Aⁿ = cn−1Aⁿ⁻¹ B_n−1· Aⁿ = Aⁿ

,

O = Aⁿ+ cn−1Aⁿ⁻¹+··· + c1A + c0In=χA(A).

 β V ordered basis, T : V→ V linear operator, χT(x) =χ[T ]_β(x).

χT(T ) linear operator, β representative matrix , Lemma 3.2.4 [χ[T ]_β(T )]_β=χ[T ]_β([T ]_β).

Theorem 3.2.10 [χT(T )]_β = O, Lemma 3.1.1 χT(T ) = O.

linear operator Cayley-Hamilton Theorem.

Corollary 3.2.11 (Cayley-Hamilton Theorem). V finite dimensional F-space, T : V → V linear operator, χT(T ) = O.

3.3. Minimal Polynomial

A n× n matrix, A characteristic polynomial, 數 n

f (X )∈ F[x] f (A) = O. 數 ?

, A = I_n , χIn(x) = (x− 1)ⁿ, f (x) = x− 1, f (I_n) = I_n− In= O.

數 f (X )∈ F[x] f (A) = O.

Definition 3.3.1. A∈ Mn(F), f (x)∈ F[x] f (A) = O, 數 monic polynomial ( 數 1) A minimal polynomial, µA(x) .

數 f (x)∈ F[x] f (A) = O, monic

性. f (x), g(x)∈ F[x] 數 monic polynomial

f (A) = g(A) = O, 數 deg( f ) = deg(g), f (x), g(x) monic, deg( f (x)− g(x)) < deg( f (x)). f (A)− g(A) = O − O = O, 數

f (x)− g(x) , f (x) = g(x), minimal polynomialµA(x) . A∼ B, minimal polynomial µA(x),µB(x) . minimal polynomial 性 .

Lemma 3.3.2. A∈ Mn(F) f (x)∈ F[x]. f (A) = O µA(x)| f (x).

Proof. f (x)|µA(x), h(x)∈ F[x] f (x) =µA(x)h(x), µA(A) = O,

Lemma 3.2.1 f (A) =µA(A)·h(A) = O·h(A). ,

f (A) = O.

, F field, f (x) =µA(x)h(x) + r(x), h(x), r(x)∈ F[x] deg(r(x)) < deg(µA(x)). f (A) = O

O = f (A) =µA(A)· h(A) + r(A) = O · h(A) + r(A) = r(A).

r(x)∈ F[x] 數 µA(x) r(A) = O . µA(x) A minimal

polynomial r(x) , f (x) µA(x) , µA(x)| f (x). 

A∼ B, Lemma 3.2.2 µA(B)∼µA(A) = O, similar

( invertible matrix P, P⁻¹·O·P = O), µA(B) = O. Lemma 3.3.2 µB(x)|µA(x). µB(A)∼µB(B) = O, µA(x)|µB(x). µA(x),µB(x) monic,

µA(x) =µB(x). .

Proposition 3.3.3. A, B∈ Mn(F) A∼ B, µA(x) =µB(x).

linear operator minimal polynomial.

Definition 3.3.4. V finite dimensional F-space, T : V→V linear operator.

f (x)∈ F[x] f (T ) = O, 數 monic polynomial T

minimal polynomial, µT(x) .

matrix , T minimal polynomial . Lemma 3.3.2

( 數 數 數) .

Lemma 3.3.5. V finite dimensional F-space, T : V→V linear operator.

f (T ) = O µT(x)| f (x).

Question 3.5. Lemma 3.3.5 ?

β V ordered basis, T characteristic polynomialχT(x) T representative matrix [T ]_β characteristic polynomial χ[T ]_β(x) . T minimal polynomial

µT(x) µ[T ]_β , 探 .

3.3. Minimal Polynomial 51

Proposition 3.3.6. V finite dimensional F-space, β V ordered basis T : V → V linear operator.

µT(x) =µ[T ]_β(x).

Proof. , f (x)∈ F[x], Lemma 3.2.4 Lemma 3.1.1 f (T ) = O⇔ [ f (T)]_β= O⇔ f ([T]_β) = O.

µT(T ) = O µT([T ]_β) = O, Lemma 3.3.2 µ[T ]_β(x)|µT(x).

µ[T ]_β([T ]_β) = O, µ[T ]_β(T ) = O, µT(x)|µ[T ]_β(x). µT(x),µ[T ]_β(x) monic

polynomial, µT(x) =µ[T ]_β(x). 

探 minimal polynomial characteristic polynomial . Theorem 3.3.7.

(1) A∈ Mn(F), µA(x)|χA(x). λ ∈ F χA(λ) = 0 µA(λ) = 0.

(2) V finite dimensional F-space, T : V→V linear operator, µT(x)|χT(x).

λ ∈ F χT(λ) = 0 µT(λ) = 0.

Proof.

(1) χA(A) = O, Lemma 3.3.2 µA(x)|χA(x). µA(λ) = 0 χA(λ) = 0.

, χA(λ) = 0, det(λIn− A) = 0 λIn− A invertible matrix.

µA(x) x−λ, µA(x) = (x−λ)h(x) + r, h(x)∈ F[x] r∈ F. 代 A, O =µA(A) = (A−λIn)· h(A) + rIn. r̸= 0, (λIn− A) · h(A) = rIn

(λIn− A) · r⁻¹h(A) = In. 代 r⁻¹h(A) λIn− A inverse, λIn− A invertible matrix , r = 0, x−λ | µA(x). µA(λ) = 0.

(2) linear operator T : V→V, V ordered basisβ, χT(x) =χ[T ]_β(x) µT(x) =µ[T ]_β(x). (1) [T ]_β, µT(x)|χT(x)

χT(λ) = 0 ⇔ µT(λ) = 0.

 Example 3.3.8. 前 Example 3.2.6 characteristic polynomial

minimal polynomial. χA1(x) = x², Theorem 3.3.7 µA1(x) x x². A₁̸= O, A1 minimal polynomial x, µA1(x) = x².

χA2(x) = x²−1, Theorem 3.3.7 x−1 x + 1 µA2(x) , µA2(x)| x²−1 µA2(x) = x²− 1.

χA3(x) = x²+ 1, Theorem 3.3.7 µA3(x)| x²+ 1. F =R, x²+ 1 monic factor ( ) 1 x²+ 1, minimal polynomial 數 , µA3(x)| x²+ 1.

F =C, i,−i x²+ 1 = 0 , Theorem 3.3.7 µA3(x) = x²+ 1.

Question 3.6. A∈ M2(R), µA(x)̸=χA(x) ?

Question 3.7. A∈ Mn(F) χA(x) = (x−λ1)···(x −λn) λi∈ F λi̸=λj for i̸= j, µA(x) ?

Theorem 3.3.7 , 學 代 數.

p(x)∈ F[x] irreducible polynomial, F finite extension ˜F,

p(x) = 0 F˜ . λ ∈ ˜F ( p(λ) = 0), f (x)∈ F[x],

f (λ) = 0, p(x) irreducible, p(x)| f (x). A∈ Mn(F), A Mn( ˜F) .

A characteristic polynomial A , det(xI_n− A),

A M_n(F) M_n( ˜F) matrix . minimal polynomial field

. A Mn( ˜F) , minimal polynomial ( µ˜A(x) ),

F[x]˜ 數 monic polynomial f (x) f (A) = O. µA(x)∈ F[x] ⊆ ˜F[x],

Lemma 3.3.2 µ˜A(x)|µA(x). , .

Theorem 3.3.9.

(1) A∈ Mn(F) p(x)∈ F[x] irreducible polynomial. p(x)|χA(x) p(x)|µA(x).

(2) V finite dimensional F-space, T : V→ V linear operator, p(x)∈ F[x]

irreducible polynomial. p(x)|χT(x) p(x)|µT(x).

Proof.

(1) Theorem 3.3.7 µA(x)|χA(x), p(x)|µA(x) p(x)|χA(x).

, p(x)∈ F[x] irreducible p(x)|χA(x). F˜ F finite extension,

p(x) = 0 F˜ λ. A M_n( ˜F) µ˜A(x)∈ ˜F[x]

A∈ Mn( ˜F) F[x]˜ minimal polynomial. p(x)|χA(x), χA(λ) = 0.

Theorem 3.3.7 F˜ , µ˜A(λ) = 0. µ˜A(x)|µA(x), µA(λ) = 0. µA(x)∈ F[x] p(x)∈ F[x] irreducible, p(x)|µA(x).

(2) linear operator T : V→V, V ordered basisβ, χT(x) =χ[T ]_β(x) µT(x) =µ[T ]_β(x). (1) [T ]_β,

p(x)|χT(x)⇔ p(x) |χ[T ]_β(x)⇔ p(x) |µ[T ]_β(x)⇔ p(x) |µT(x).

 Question 3.8. A∈ Mn(F) χA(x) = p^c₁¹(x)··· p^c_k^k(x) c_i∈ N, pi(x)∈ F[x] monic irreducible polynomial p_i(x)̸= pj(x) for i̸= j, µA(x) ?

Example 3.3.10. linear operator T : P2(R) → P2(R)

T (1) = 2x²− 1,T(x + 1) = 3x²+ 2x + 2, T (−x²+ x + 1) = 4x²+ 2x + 2.

T minimal polynomialµT(x).

P₂(R) ordered basisβ = (−x²+ x + 1, x + 1, 1).

3.4. Internal Direct Sum 53

T (−x²+ x + 1) = (−4)(−x²+ x + 1) + 6(x + 1) T (x + 1) = (−3)(−x²+ x + 1) + 5(x + 1)

T (1) = (−2)(−x²+ x + 1) + 2(x + 1) + (−1)1 [T ]_β=



 −4 −3 −2

6 5 2

0 0 −1



. χT(x) =χ[T ]_β(x) = (x + 1)²(x− 2).

([T ]_β+ I₃)· ([T]_β− 2I3) =



 −3 −3 −2

6 6 2

0 0 0



 ·



 −6 −3 −2

6 3 2

0 0 −3



 =



 0 0 6

0 0 −6

0 0 0



,

µT(x) =µ[T ]_β(x)̸= (x + 1)(x − 2), µT(x) =µ[T ]_β(x) = (x + 1)²(x− 2).

([T ]_β+ I₃)²· ([T]_β− 2I3) =



 −3 −3 −2

6 6 2

0 0 0



 ·



 0 0 6

0 0 −6

0 0 0



 = O.

Question 3.9. ordered basis (x², x, 1) Question 3.3.10. ? 3.4. Internal Direct Sum

linear operator T : V → V, ordered basis, T representative

matrix matrix. V internal direct sum of

T -invariant subspaces. linear operator, 探 internal direct sum

性 .

Chapter 1 direct sum external direct sum,

vector space , vector space. vector space ,

探 internal direct sum .

U,W V subspace. 數 T : U⊕W → U +W,

T ((u, w)) = u + w, ∀u ∈ U,w ∈ W.

T well-defined function, T onto linear transforma- tion. Ker(T ) ? (u, w)∈ Ker(T), T ((u, w)) = u + w = OV, u =−w. u∈U,w ∈W, u =−w ∈U ∩W. , u∈U ∩W, (u,−u) ∈U ⊕W,

T ((u,−u)) = OV. Ker(T ) ={(u,−u) | u ∈ U ∩W}.

Question 3.10. T : U⊕W → U +W 數 U,W V subspace

?

Question 3.11. {(u,−u) | u ∈ U ∩W} ≃ U ∩W. the First Isomorphism Theo-

rem, (U⊕W)/(U ∩W) ≃ U +W?

, U∩W = {OV} , (OV, OV) = OU⊕W, Ker(T ) = OU⊕W. T

one-to-one, .

Proposition 3.4.1. U,W V subspace, U∩W = {OV}, U⊕W ≃ U +W.

, U,W V subspace U∩W = {OV} , U + W

U⊕W . U⊕W V subspace U +W , 前

vector space. U⊕W U∩W = {OV}. ,

U,W internal direct sum. , U,W V subspace U∩W ̸= {OV}

U⊕W 代 external direct sum. U,W V subspace,

U⊕W ⊆ V internal direct sum, U∩W = {OV}. U,W

, U⊕W external direct sum.

V finite dimensional vector space, U V subspace.

V subspace W V = U⊕W. U basis S ={u1, . . . , u_m}, S 大 V basis{u1, . . . , um, w1, . . . , wn}. W = Span({w1, . . . , wn}), {u1, . . . , u_m, w₁, . . . , w_n} linearly independent, U∩W = {OV}. U⊕W

U,W internal direct sum. V = Span({u1, . . . , u_m, w₁, . . . , w_n}),

U⊕W = V. linearly independent basis ,

V subspace U, V U⊕W W .

Example 3.4.2. F²={(x,y) | x,y ∈ F}, U ={(x,0) | x ∈ F}, W₁={(0,y) | y ∈ F}

W₂={(y,y) | y ∈ F} F²= U⊕W1 F²= U⊕W2.

V internal direct sum V = U⊕W v∈ V, u∈ U

w∈ W v = u + w. V subspaces internal direct sum 性

.

Proposition 3.4.3. U,W V subspaces.

(1) V = U⊕W.

(2) v∈ V, u∈ U,w ∈ W v = u + w.

(3) U,W basis S₁, S₂, S₁∩ S2= /0 S₁∪ S2 V basis.

Proof. (1)⇒ (2): V = U +W , v∈ V, u∈ U,w ∈ W v = u + w.

u^′∈ U,w^′∈ W v + w = u^′+ w^′, u−u^′= w^′−w ∈ U ∩W = {OV}, u = u^′ w = w^′.

(2)⇒ (3): v∈ S1∩ S2, v∈ U ∩W. v = v + O_V = O_V + v v

U , v W O_V W , O_V U ,

性 v = O_V. v∈ S1, S₁ linearly independent , S₁∩ S2= /0.

v∈ V, u∈ U,w ∈ W v = u + w, S₁, S₂ U,W basis,

u₁, . . . , um∈ S1, w1, . . . , wn∈ S2 c1, . . . , cm, d1, . . . , dn∈ F u = c₁u₁+··· + cmu_m, w = d₁w₁+··· + dnw_n. v = c₁u₁+··· + cmu_m+ d₁w₁+··· + dnw_n, S₁∪ S2 V

3.4. Internal Direct Sum 55

spanning set. S₁∪ S2 linearly independent, Corollary 1.4.4 v̸= OV v∈ Span(S1)∩ Span(S2) = U∩W. 前 S1∩ S2= /0 , v

U , W 性 . S1∪ S2 linearly independent.

(3)⇒ (1): S1∪ S2 V basis, V = Span(S1) + Span(S2) = U +W .

U∩W = {OV}. S₁∩S2= /0 (S₁∪S2)\S1= S₂, Corollary 1.4.4 S₁∪S2

linearly independent Span(S1)∩ Span(S2) ={OV}, U∩W = {OV}.  subspaces internal direct sum subspaces internal

direct sum. V = U⊕W, W W subspaces W1,W2 direct

sum, W = W₁⊕W2, V = U⊕W1⊕W2. W = W₁⊕W2, W₁∩W2={OV}, V = U⊕W, U∩W1⊆ U ∩W = {OV}, U ∩W2⊆ U ∩W = {OV}.

( W₁∩W2={OV}, U ∩W1={OV} U∩W2={OV}) Proposition 3.4.3 性 ( v u∈ U,w1∈ W1, w₂∈ W2 v = u + w₁+ w₂),

.

Example 3.4.4. Example 3.4.2 U∩ W1 = W₁∩ W2 = U∩ W2={(0,0)}, (x, y)∈ F², y̸= 0, (x, y) = (x, 0) + (0, y) + (0, 0) = (x− y,0) + (0,0) + (y,y), ((0, 0)∈ W1 (0, 0)̸= (0,y) ∈ W1. , (0, 0)̸= (y,y) ∈ W2. F²

U,W₁,W₂ .

internal direct sum ? external direct sum . V₁,V₂,V₃ V subspace, external direct sum V₁⊕V2⊕V3 V₁+ V₂+ V₃ linear transformation T , T (v1, v2, v3) = v1+ v2+ v3. T onto. T one- to-one, Ker(T ) ={(OV, O_V, O_V)} v₁∈ V1, v₂∈ V2, v₃∈ V3 v₁+ v₂+ v₃= O_V,

v₁= v₂= v₃= O_V. v₁+ v₂+ v₃= O_V, v₁=−(v2+ v₃)∈ V1∩ (V2+ V₃),

v₂∈ V2∩(V1+V3), v3∈ V3∩(V1+V2). V1∩(V2+V3) = V2∩(V1+V3) = V3∩(V1+V2) = {OV}, Ker(T ) ={(OV, O_V, O_V)}. , v₁∈V1∩(V2+V₃), v₁∈V2, v₃∈V3

v₁= v2+ v3, (v1,−v2,−v3)∈ Ker(T). Ker(T ) ={(OV, OV, OV)} v₁= OV, V₁∩ (V2+ V₃) = O_V. V₂∩ (V1+ V₃) = V₃∩ (V1+ V₂) = O_V.

subspaces, .

Definition 3.4.5. V1, . . . ,Vk V subspaces, V_i∩ (

∑

j̸=i

V_j) ={OV}, ∀i = 1,...,k

V subspace V1+··· +Vk V1, . . . ,V_k internal direct sum, V1⊕ ··· ⊕Vk . 再 , V subspaces V1, . . . ,V_k, V1+··· +Vk subspace.

V₁⊕···⊕Vk⊆ V internal direct sum, V₁, . . . ,V_k V_i∩(∑j̸=iV_j) =

{OV}, ∀i = 1,...,k . , decomposition theorem,

vector space subspaces internal direct sum, 再 external direct

sum, 再 internal direct sum.

vector space subspaces direct sum, subspaces direct sum

性 . Proposition 3.4.3 , 再 .

Proposition 3.4.6. V1, . . . ,Vk V subspace.

(1) V = V1⊕ ··· ⊕Vk.

(2) v∈ V, i = 1, . . . , k v_i∈ Vi v = v₁+··· + vk. (3) Vi basis Si, S1∩ ··· ∩ Sk= /0 S1∪ ··· ∪ Sk V basis.

Question 3.12. V finite dimensional vector space V1, . . . ,V_k V subspaces V = V₁⊕ ··· ⊕Vk, dim(V ) dim(V₁) +··· + dim(Vk) ?

U,W V subspaces V = U⊕W, W₁, . . . ,W_k W subspaces W =

W1⊕ ··· ⊕ Wk, V = U⊕ W1⊕ ··· ⊕ Wk ? .

v∈ V, V = U⊕W u∈ U,w ∈ W v = u + w. W = W1⊕ ··· ⊕Wk, w_i ∈ Wi, w = w₁+··· + wk. v∈ V, u∈ U,w1 ∈ W1, . . . , wk∈ Wk v = u + w₁+···+wk ( 性). u^′∈ U,w^′₁∈ W1, . . . , wk∈ Wk

v = u^′+ w^′₁+··· + w^′_k, u, u^′∈ U w₁+··· + wk, w^′₁+··· + w^′_k∈ W, V = U⊕W u = u^′ w₁+···+wk= w^′₁+···+w^′_k. w_i, w^′_i∈ Wi, W = W1⊕···⊕Wk w_i= w^′_i

( 性), Proposition 3.4.6 .

Corollary 3.4.7. U,W V subspaces V = U⊕ W, W₁, . . . ,W_k W subspaces W = W1⊕ ··· ⊕Wk, V = U⊕W1⊕ ··· ⊕Wk.

3.5. Primary Decomposition

linear operator. T : V→ V linear operator, V

subspaces direct sum, subspaces ordered basis V ordered basis

T representative matrix . , T

subspaces ( linear operator), .

Definition 3.5.1. T : V → V linear operator. W V subspace T (W )⊆ W ( w∈ W T (w)∈ W), W T -invariant.

Question 3.13. T : V→V linear operator. subspaces T -invariant?

(1) V. (2){OV}. (3) Im(T ). (4) Ker(T ).

, T : V → V linear operator, f (x) = a_dx^d+··· + a1x + a₀∈ F[x], linear operator f (T ) = adT^◦d+··· + a1T + a0id.

Lemma 3.5.2. V F-space, T : V → V linear operator. W T -invariant, f (x)∈ F[x], W f (T )-invariant

3.5. Primary Decomposition 57

Proof. W T -invariant, w∈ W, T (w)∈ W T^◦2(w) = T (T (w))∈ W.

數學 T^◦i(w)∈ W, ∀i ∈ N. f (x) = adx^d+··· + a1x + a0∈ F[x], W subspace, f (T )(w) = a_dT^◦d(w) +··· + a1T (w) + a0w∈ W, ∀w ∈ W. W f (T )-

invariant. 

Im(T ) Ker(T ) T -invariant. f (x)∈ F[x]

T -invariant subspaces.

Lemma 3.5.3. V F-space, T : V→ V linear operator f (x)∈ F[x]. Im( f (T )) Ker( f (T )) T -invariant subspaces.

Proof. w∈ Im( f (T)), v∈ V w = f (T )(v). Lemma 3.2.3 T◦ f (T) = f (T) ◦ T,

T (w) = T ( f (T )(v)) = (T◦ f (T))(v) = ( f (T) ◦ T)(v) = f (T)(T(v)) ∈ Im( f (T)), Im( f (T )) T -invariant.

v∈ Ker( f (T)), f (T )(v) = OV. f (T )(T (v)) = T ( f (T )(v)) = T (OV) = OV,

T (v)∈ Ker( f (T)), Ker( f (T )) T -invariant. 

linear operator T : V→ V, V subspace W , T

W , T|W: W → V, T|W(w) = T (w),∀w ∈ W. W V

linear transformation, the restriction on W . W T -invariant , T (w)∈ W,

∀w ∈ W, T|W : W → W, W linear operator. 探 T|W

T minimal polynomial . f (x)∈ F[x], W f (T )-invariant (Lemma 3.5.2), f (T )|W f (T|W) W linear operator

. w∈ W,

T^◦2|W(w) = T^◦2(w) = T (T (w)) = T|W(T|W(w)) = T|W◦2(w),

T^◦2|W T|W◦2 W linear operator. 數學 T^◦i|W =

T|W◦i,∀i ∈ N. f (x) = a_dx^d+··· + a1x + a₀∈ F[x], w∈ W, f (T )|W(w) = f (T )(w) = a_dT^◦d|W(w) +··· + a1T|W(w) + a₀id|W(w)

= a_dT|W◦d(w) +··· + a1T|W(w) + a0id|W(w) = f (T|W)(w).

f (T )|W f (T|W) W linear operator,

f (T )|W = f (T|W). (3.3)

, Lemma.

Lemma 3.5.4. T : V → V linear operator, W T -invariant subspace, T restriction on W , T|W: W→ W W linear operator, minimal polynomialµT|W(x)

µT|W(x)|µT(x).