大學線性代數初步

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大學線性代數初步

大學數學

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數學大學線性代數 , 大

數 .

大學線性代數學 ,

學線性代數 . 大學線性代數

. 學大大 ( ) 線性

代數 , . 大學線性代數 , 大

學數學 . 大大數學

, 學 , 線性代數

. , 線性代數

數學 . , .

, .

學 , 數學學 , 線性

代數 . 線性代數 .

, . 學

. , (Question).

, 大

. , 線性代數 . ,

學線性代數學線性代數 , .

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代. , .

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v

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Chapter 9

Eigenvalues and Eigenvectors

, n× n matrices eigenvalue eigenvector.

eigenvalue eigenvector 性 . n×n

matrix .

9.1. Characteristic Polynomial

n× n matrix A, v∈ Rⁿ, 數學 A^kv k 數

. Fibonacci sequence F1, F2, . . . . Fk+1= Fk+ Fk−1 數 . A =

[1 1 1 0 ]

k≥ 2 v_k= [ F_k

F_k₋₁ ]

,

Avk= [1 1

1 0 ][ Fk

F_k−1 ]

=

[Fk+ Fk−1

F_k ]

= [Fk+1

F_k ]

= vk+1.

v₃= Av₂, v4= Av₃= A(Av₂) = A²v₂, . . . , v_k+1= A^k−1v₂.

k≥ 2, A^k⁻¹v₂, F_k+1 .

k 大 , A^kv . , λ ∈ R

Av =λv , A²v = A(Av) = A(λv) = λ(Av) = λ²v. A³v =λ³v, . . . ,

A^kv =λ^kv. , A^kv. v

λ ∈ R Av =λv . .

Deﬁnition 9.1.1. A n× n matrix. v∈ Rⁿ, λ ∈ R

Av =λv, v A eigenvector, λ A eigenvalue.

, A eigenvector . v A eigenvector,

λ,λ^′∈ R Av =λv = λ^′v, (λ − λ^′)v = O v̸= O, λ = λ^′. A

eigenvector v 數 λ Av =λv. eigenvector v

eigenvalue λ.

189

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Question 9.1. A n× n matrix, v ∈ Rⁿ Av = O. v

eigenvector? eigenvalue ?

n× n matrix eigenvector eigenvalue ?

eigenvalue, eigenvector. λ ∈ R A

eigenvalue, v∈ Rⁿ Av =λv. I_nv = v,

λv = (λIn)v. Av =λv (A−λIn)v = O. 言 ,λ A eigenvalue

n× n matrix A −λIn linear system (A−λIn)x = O nontrivial solution x = v.

Theorem 3.5.9, A−λIn invertible, Theorem 8.2.6(1)

det(A−λIn) = 0. 言 , A eigenvalueλ λ det(A−λIn) = 0.

λ det(A−λIn) = 0 ? A = [ai j], t 數, det(A−tIn).

A−tIn=







a_{1 1}−t a_{1 2} ··· a_{1 n} a2 1 a2 2−t ··· a2 n

... ... . .. ... a_{n 1} a_{n 2} ··· an n−t







數學 , det(A−tIn) t 數 n 數 .

t =λ 數 , λ det(A−λIn) = 0, λ A

eigenvalue. , λ A eigenvalue, t =λ det(A−tIn)

. det(A−tIn) A eigenvalue,

.

Deﬁnition 9.1.2. A n× n matrix, t 數 p_A(t) = det(A−tIn).

pA(t) A characteristic polynomial ( )..

t =λ characteristic polynomial pA(t)

λ A eigenvalue. A Rⁿ eigenvectors,

t =τ p_A(t) ( 數 ), v∈ Rⁿ Av =τv.

Av∈ Rⁿ, τv ̸∈ Rⁿ, Av =τv . , eigenvalue

characteristic polynomial .

Example 9.1.3. A =



 2 0 1 1 3 1 1 1 2



, A characteristic polynomial pA(t) =

det(A−tI3) = det



 2−t 0 1

1 3−t 1

1 1 2−t



. row

pA(t) = (2−t)det

[ 3−t 1 1 2−t

] + det

[ 1 3−t

1 1

]

= (2−t)(t²− 5t + 6 − 1) + (1 − (3 −t)).

pA(t) = (2−t)(t²− 5t + 4) = (2 −t)(t − 1)(t − 4). t = 1, 2, 4 A characteristic polynomial , A eigenvalues 1, 2, 4.

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9.1. Characteristic Polynomial 191

A n×n matrix , characteristic polynomial det(A−tIn)

t . determinant , . 數學

column

row. 2×2 matrix

[ a b c d

]

characteristic polynomial det

[ a−t b c d−t

]

t (a− t)(d − t) bc

數 , t² t 數 (a−t)(d −t) t² t 數

at²− (a + d)t . 3× 3 matrix A =



 a b c d e f g h i



 characteristic polynomial.

row det



 a−t b c

d e−t f

g h i−t



 = (a −t)det[

e−t f h i−t

]

− bdet

[ d f g i−t

] + c det

[ d e−t

g h

] .

前 2× 2 det

[ e−t f h i−t

]

t² t 數

(e− t)(i − t) t² t 數 , (a− t)(e − t)(i − t) t³ t² 數.

det

[ d f g i−t

] det

[ d e−t

g h

]

t , det(A− tI3) t³ t²

數 (a−t)(e −t)(i −t) . A chacteristic polynomial p_A(t)

3 (−1)³t³+ (−1)²(a + e + i). a, e, i A diagonal

entries, a + e + i A trace, tr(A) . 數學 ,

A = [ai j] n× n matrix , A characteristic polynomial pA(t) = det(A−tIn)

t n 數 , (a1 1−t)(a2 2−t)···(an n−t)

(−1)ⁿtⁿ+ (−1)ⁿ⁻¹(a_{1 1}+··· + an n)tⁿ⁻¹. A diagonal entries a_{1 1}+··· + an n

tr(A), .

Proposition 9.1.4. A n×n matrix. A characteristic polynomial t n 數 . tⁿ 數 (−1)ⁿ, tⁿ⁻¹ 數 (−1)ⁿ⁻¹tr(A) 數數 det(A).

Proof. p_A(t) = det(A−tIn), 前 p_A(t) 數 . p_A(t)

數 pA(0) = det(A− 0In) = det(A).

Question 9.2. A n× n matrix. A eigenvalues?

eigenvalue . λ ∈ R A eigenvalue. t =λ

A characteristic polynomial pA(t) = det(A−tIn) . t−λ p_A(t). (t−λ)^m p_A(t), (t−λ)^m+1 p_A(t), eigenvalueλ

algebraic multiplicity (代數 數) m. t =λ pA(t) , λ

algebraic multiplicity 1.

Question 9.3. Identity matrix I_n eigenvalue ? algebraic multiplicity ?

characteristic polynomial 性 . n× n

matrices characteristic polynomial . ,

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characteristic polynomial . A, B n× n matrices, n× n invertible matrix U, B = U⁻¹AU , A, B similar (

). A B characteristic polynomial .

Proposition 9.1.5. A, B n× n matrices n× n invertible matrix U B = U⁻¹AU . A B characteristic polynomial.

Proof. B characteristic polynomial det(B−tIn) = det(U⁻¹AU−tIn).

性

U⁻¹(A−tIn)U = U⁻¹AU−U⁻¹(tI_n)U = U⁻¹AU−tU⁻¹InU = U⁻¹AU−tIn. determinant 性 (Theorem 8.2.6)

det(B−tIn) = det(U⁻¹(A−tIn)U ) = det(U⁻¹) det(A−tIn) det(U ) = det(A−tIn).

A B characteristic polynomial.

characteristic polynomial A A^T characteristic polynomial.

Proposition 9.1.6. A n×n matrix, A A^T characteristic polynomial Proof. trnaspose 性 (A− tIn)^T = A^T− tIn^T= A^T− tIn (Proposition 3.2.4),

Theorem 8.2.6 (3),

P_A^T(t) = det(A^T−tIn) = det((A−tIn)^T) = det(A−tIn) = PA(t).

Question 9.4. A A^T eigenvalues eigenvalue A A^T

algebraic multiplicity . 9.2. Eigenspace

n× n matrix eigenvalue , eigenvalue eigenvectors.

A n× n matrix λ ∈ R A eigenvalue. det(A−λIn) = 0, (A−λIn)x = O nontrivial solution. v∈ Rⁿ

x = v (A−λIn)x = O . v (A−λIn)v = O, Av =λv.

v A λ eigenvalue eigenvector. , v A λ eigenvalue

eigenvector, x = v (A−λIn)x = O nontrivial solution.

n× n matrix A −λIn nullspace ( {v ∈ Rⁿ| (A −λIn)v = O}) A

λ eigenvector. nullspace vector space, .

Deﬁnition 9.2.1. A n× n matrix λ ∈ R A eigenvalue. A−λIn

nullspace A eigenvalueλ eigenspace. E_A(λ) .

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9.2. Eigenspace 193

λ eigenspace λ eigenvalue eigenvectors .

O eigenvector, vector space O. λ eigenspace

λ eigenvalue eigenvectors O . vector space

? vector space 性, vector space dimension

大 . EA(λ) dimension eigenvalue λ geometric multiplicity

( 數). eigenvalue λ algebraic multiplicity λ

eigenvectors , λ geometric multiplicity .

Example 9.2.2. A =



 −1 4 2

−1 3 1

−1 2 2



, B =



 0 3 1

−1 3 1

0 1 1



. A B

characteristic polynomial−(t − 1)²(t− 2). 1 2 A B eigenvalues.

A B, eigenvalue 1 algebraic multiplicity 2, eigenvalue 2 algebraic

multiplicity 1. A B eigenspace.

A eigenvalue 1 eigenspace, A− I3 =



 −2 4 2

−1 2 1





null space. elementary row operations, echelon form



 1 −2 −1

0 0 0



.

EA(1) = Span(



2 1 0



,



1 0 1



). A eigenvalue 1 eigenvector



2 1 0







1 0 1



 linear combination nonzero vector. v =



4 1 2



 =



2 1 0



 + 2



1 0 1





Av =



 −1 4 2

−1 3 1

−1 2 2







4 1 2



 =



4 1 2



 = v.

dim(EA(1)) = 2, A eigenvalue 1 geometric multiplicity 2.

A eigenvalue 2 eigenspace, A− 2I3 =



 −3 4 2

−1 1 1

−1 2 0



 null space.

elementary row operations, echelon form



 1 −2 0

0 1 −1

0 0 0



. EA(2) =

Span(



2 1 1



). A eigenvalue 2 eigenvector



2 1 1



 nonzero vector, A eigenvalue 2 geometric multiplicity 1.

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B eigenvalue 1 eigenspace, B− I3=



 −1 3 1

−1 2 1 0 1 0



 null space.

elementary row operations, echelon form



 1 0 −1

0 1 0

0 0 0



. EB(1) = Span(



1 0 1



).

B eigenvalue 1 eigenvector



1 0 1



 nonzero vector,

B eigenvalue 1 geometric multiplicity 1. B eigenvalue 2 eigenspace, B− 2I3=



 −2 3 1

−1 1 1

0 1 −1



 null space. elementary row

operations, echelon form



 1 0 −2 0 1 −1

0 0 0



. EA(2) = Span(



2 1 1



). A

eigenvalue 2 eigenvector



2 1 1



 nonzero vector, A

eigenvalue 2 geometric multiplicity 1.

Example 9.2.2 characteristic polynomial,

eigenvectors 大 , eigenspace dimension .

eigenvalues algebraic multiplicity , geometric multiplicity .

Question 9.5. Identity matrix I_n eigenvalue geometric multiplicity ?

Proposition 9.1.6 A A^T characteristic polynomial eigenvalue eigenvalue A A^T algebraic multiplicity .

geometric multiplicity , .

Proposition 9.2.3. A n× n matrix λ ∈ R A eigenvalue. λ A geometric multiplicity λ A^T geometric multiplicity .

Proof. dim(EA(λ)) = dim(EA^T(λ)), dim(N(A−λIn)) = dim(N(A^T−λIn)).

Theorem 4.4.13 dim(N(A−λIn)) = null(A−λIn) = n−rank(A−λIn), A∈ Mn×n

dim(N(A^T−λIn)) = n−rank(A^T−λIn). A^T−λIn= (A−λIn)^T rank((A−λIn)^T) = rank(A−λIn) (Proposition 4.4.14), dim(N(A−λIn)) = dim(N(A^T−λIn)).

v A eigenvector, eigenvalue λ, v nonzero vector

eigenvalue λ eigenvector. v, w A eigenvectors

eigenvalue , v, w . v, w linearly independent.

.

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9.3. Diagonalization 195

Proposition 9.2.4. A n×n matrix v₁, . . . , v_k A eigenvectors. v₁, . . . , v_k eigenvalues , v₁, . . . , v_k linearly independent.

Proof. 數學 . 前 k = 2 , k− 1

eigenvectors . k eigenvectors . v₁, . . . , v_k A

eigenvectors eigenvalue λ1, . . . ,λk ( Avi=λiv_i, for i = 1, . . . , n).

v₁, . . . , v_k₋₁ linearly independent. , v₁, . . . , v_k₋₁, v_k linearly dependent. Lemma 4.2.4, v_k ∈ Span(v1, . . . , v_k₋₁). c1, . . . , c_k₋₁∈ R

v_k= c₁v₁+··· + ck−1v_k₋₁ (9.1) eigenvector

λkv_k= Avk= A(c1v₁+··· + ck−1v_k₋₁) = c1Av1+··· + ck−1Avk−1= c1λ1v₁+···ck−1λk−1v_k₋₁. (9.2)

(9.1) λk (9.2)

c₁(λk−λ1)v₁+··· + ck−1(λk−λk−1)v_k₋₁= O. (9.3)

v_k ̸= O, c1, . . . , c_k₋₁ 0. eigenvalue , i =

1, . . . , k− 1, λk−λi̸= 0. c1(λk−λ1), . . . , c_k−1(λk−λk−1) 0 數.

, (9.3) v₁, . . . , v_k₋₁ linearly dependent, ,

.

9.3. Diagonalization

A n× n matrix, v A eigenvector eigenvalue

λ, A^kv =λ^kv. v A eigenvector ?

, A^kv . , v₁, . . . , v∈ Rⁿ Rⁿ

basis A eigenvectors , A^kv.

v∈ Rⁿ, basis c1, . . . , cn∈ R v = c₁v₁+··· + cnv_n. v₁, . . . , vn

eigenvalues λ1, . . . ,λn,

Av = A(c1v₁+··· + cnv_n) = c₁Av1+··· + cnAvn= c₁λ1v₁+··· + cnλnv_n. A

A²v = A(Av) = A(c₁λ1v₁+··· + cnλnv_n) = c₁λ1Av₁+··· + cnλnAv_n= c₁λ1²v₁+··· + cnλn²v_n. 數學

A^kv = c₁λ1^kv₁+··· + cnλn^kv_n.

matrix .

Deﬁnition 9.3.1. A n× n matrix. Rⁿ basis v1, . . . , vn v_i A eigenvectors, A diagonalizable ( ).

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diagonalizable ? v₁, . . . , v∈ Rⁿ Rⁿ basis A eigenvectors, eigenvalues λ1, . . . ,λn. Av1=λ1v₁, . . . , Av_n= λnv_n.

A



 v ₁ v₂ ··· vn



 =





Av1 Av2 ··· Avn



 =





λ1v₁ λ2v₂ ··· λnv_n



.

(i, i)-th entry λi n× n diagonal matrix ( 線 i λi

線 0),



 v₁ v₂ ··· vn



D =



 v₁ v₂ ··· vn











λ1 0 ··· 0 0 λ2 ··· 0 ... ... . .. ...

0 0 0 λn





=





λ1v₁ λ2v₂ ··· λ3v_n



.

C =



 v ₁ v₂ ··· vn



, AC = CD. C column linearly

independent n column, C rank n, C n×n matrix C

invertible ( Theorem 3.5.2). AC = CD D = C⁻¹AC. ,

n× n invertible matrix C⁻¹AC diagonal matrix D, C n× n invertible

matrix, C n column vectors Rⁿ basis. AC = CD,

性 C i-th column A D (i, i)-th entry eigenvalue eigenvector.

C column vectors Rⁿ basis A eigenvectors, A

diagonalizable. 前 , U⁻¹AU ( U n× n invertible matrix)

matrix A similar matrix. , A diagonalizable

A diagonal matrix similar. diagonalizable .

n×n matrix A diagonalizable ? ,

eigenvectors. A eigenvectors

A diagonalizable. eigenvectors eigenvalues,

A characteristic polynomial R . λ1, . . . ,λk∈ R pA(t) = (−1)ⁿ(t−λ1)^m¹···(t −λk)^m^k. i = 1, . . . , k, mi λi

algebraic multiplicity pA(t) 數 n, m1+···+mk= n.

eigenvalue, geometric multiplicity algebraic multiplicity.

A eigenvectors , eigenvalue geometric multiplicity

algebraic multiplicity. i = 1, . . . , k,λi geometric multiplicity algebraic multiplicity, dim(EA(λi)) = m_i. v_i,1, . . . , v_i,m_i EA(λi) basis. k vectors , v_1,1, . . . , v_1,m₁, . . . , v_k,1, . . . , v_k,m_k

linearly independent. Rⁿ m₁+··· + mk = n ,

Corollary 4.3.5, Rⁿ basis. A eigenvectors,

A diagonalizable.

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9.3. Diagonalization 197

v_1,1, . . . , v_1,m₁, . . . , v_k,1, . . . , v_k,m_k linearly dependent. 0 數 c1,1, . . . , c_1,m₁, . . . , c_k,1, . . . , c_k,m_k

c1,1v_1,1+··· + c1,m1v_1,m₁+··· + ck,1v_k,1+··· + ck,mkv_k,m_k= O.

i∈ {1,...,k}, w_i = c_i,1v_i,1+··· + ci,miv_i,m_i. v_i,1, . . . , v_i,m_i linearly independent, ci,1, . . . , ci,mi 0, w_i ̸= O. wi∈ EA(λi),

w_i eigenvalue λi eigenvector. , c_{i, j}̸= 0,

i, wi eigenvalue λi eigenvectors w₁+··· + wk = O. Proposition

9.2.4 , eigenvalue eigenvectors linearly independent ,

v_1,1, . . . , v_1,m₁, . . . , v_k,1, . . . , v_k,m_k linearly independent. A

characteristic polynomial R A eigenvalue geometric

multiplicity algebraic multiplicity, A diagonalizable.

, A diagonalizable, A characteristic polynomial

R A eigenvalue geometric multiplicity algebraic

multiplicity. 前前 eigenvalue geometric

multiplicity algebraic multiplicity.

Proposition 9.3.2. A n× n matrix. λ A eigenvalue geometric multiplicity d algebraic multiplicity m, d≤ m.

Proof. dim(EA(λ)) = d, v₁, . . . , vd EA(λ) basis. v₁, . . . , vd

linearly independent, Rⁿ basis v₁, . . . , v_d, v_d+1, . . . , v_n. C

i-th column v_i n× n invertible matrix. AC = CE

E =

[ λId M1

O M2

]

. 1-st column , 數學 det(E− tIn) =

(λ −t)^ddet(M₂−tIn−d). 言 , E characteristic polynomial (t−λ)^d . A E similar ( E = C⁻¹AC), characteristic polynomial ( Proposition 9.1.5), (t−λ)^d p_A(t). λ algebraic multiplicity m,

m t−λ p_A(t) 數, d≤ m.

n× n matrix A diagonalizable. v_1,1, . . . , v_1,d₁, . . . , v_k,1, . . . , v_k,d_k Rⁿ basis, i∈ {1,...,k}, vi,1, . . . , v_i,d_i A λi eigenvalue eigenvector, λ1, . . . ,λk . v_i,1, . . . , vi,di ∈ EA(λi) linearly independent, λi

geometric multiplicity dim(E_A(λi))≥ di. λi algebraic multiplicity m_i, Proposition 9.3.2

mi≥ dim(EA(λi))≥ di,∀i = 1,...,k. (9.4) m₁+··· + mk A characteristic polynomial p_A(t) 數 ( ),

pA(t) 數 n. m1+···+mk Rⁿ dimension, n. (9.4) i = 1, . . . , k

n≥ m1+··· + mk≥ dim(EA(λi)) +··· + dim(EA(λk))≥ d1+··· + dk= n.

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“≥” “=” ( n > n ).

n = m1+··· + mk ( pA(t) 數 ) mi= dim(E_A(λi)),∀i = 1,...,k ( eigenvalue geometric multiplicity algebraic multiplicity).

.

Theorem 9.3.3. A n× n matrix. .

(1) Rⁿ basis A eigenvectors .

(2) n× n invertible matrix C C⁻¹AC diagonal matrix.

(3) A characteristic polynomial 數 A eigenvalue

geometric multiplicity algebraic multiplicity.

diagonalizable matrix , Theorem 9.3.3

diagonalizable , (3) .

Question 9.6. A n× n matrix. A diagonalizable A^T diagonalizable.

Example 9.3.4. Example 9.2.2 A =



 −1 4 2

−1 3 1

−1 2 2



, B =



 0 3 1

−1 3 1

0 1 1



.

前 characteristic polynomial−(t −1)²(t−2). A, B eigenvalue 1 algebraic multiplicity 2, eigenvalue 2 algebraic multiplicity 1.

Example 9.2.2 B eigenvalue 1 geometric multiplicity 1, B diagonalizable matrix. , A eigenvalue 1 eigenvalue 2 geometric multiplicity

algebraic multiplicity, A diagonalizable matrix. A . A eigenvalue 1 2 eigenspace E_A(1) = Span(



2 1 0



,



1 0 1



)

E_A(2) = Span(



2 1 1



),



2 1 0



,



1 0 1



,



2 1 1



 A eigenvectors R³

basis. C =



 2 1 2 1 0 1 0 1 1



 D =



 1 0 0 0 1 0 0 0 2



,

AC =



 −1 4 2

−1 3 1

−1 2 2







 2 1 2 1 0 1 0 1 1



 =



 2 1 4 1 0 2 0 1 2



 =



 2 1 2 1 0 1 0 1 1







 1 0 0 0 1 0 0 0 2



 = CD.

C invertible, C⁻¹AC = D.

, A eigenvalueλ geometric multiplicity 大 0 ( λ eigenvector ) algebraic multiplicity (Proposition 9.3.2).

λ A characteristic polynomial ( λ algebraic multiplicity 1), geometric multiplicity algebraic multiplicity ( 1).

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9.4. The Spectral Theorem 199

diagonalizable , algebraic multiplicity 1 eigenvalue

geometric multiplicity . , A characteristic polynomial R

( ), A diagonalizable.

diagonalizable, symmetric matrix. symmetric

matrix diagonalizable.

9.4. The Spectral Theorem

symmetric matrix. symmetric matrix diago-

nalizable, orthogonal diagonalizable. 數學

, , symmetric

matrix .

2× 2 symmetric matrix . A =

[ a b b c

]

, b̸= 0 (

b = 0, A diagonal matrix ). A characteristic polynomial

PA(t) = t²−(a+c)t +(ac−b²). pA(t) (a + c)²−4(ac−b²) = (a−c)²+ 4b²> 0,

P_A(t) = 0 λ1,λ2. λ1,λ2 A eigenvealue, A

diagonalizable. v₁= [ b

λ1− a ]

,

Av1= [ a b

b c

][ b λ1− a

]

=

[ λ1b b²+λ1c− ac

]

=λ1

[ b λ1− a

]

=λ1v₁.

λ1²−(a+c)λ1+ (ac−b²) = 0. b̸= 0, v₁̸= O, v₁ A eigenvector eigenvalue λ1. v₂=

[ b λ2− a

]

, v₂ A eigenvector

eigenvalue λ2. , ⟨v1, v₂⟩ = b²+λ1λ2−a(λ1+λ2) + a². 數 , λ1λ2= ac− b² λ1+λ2= a + c, ⟨v1, v₂⟩ = 0. v₁, v₂ R² basis

A eigenvectors , . diagonalizable

orthogonal diagonalizable. .

Deﬁnition 9.4.1. A∈ Mn×n, Rⁿ orthogonal basis v₁, . . . , v_n v_i A eigenvectors, A orthogonal diagonalizable.

, Deﬁnition 9.4.1 u_i= _∥v¹

i∥v_i u₁, . . . , u_n Rⁿ orthonormal basis A eigenvectors. A orthogonal diagonalizable Rⁿ

orthonormal basis A eigenvector . u_i eigenvalue λi

Q =



 u ₁ u₂ ··· un



 AQ = QD D (i, i)-th entry λi diagonal matrix,

A Q⁻¹AQ = D. eigenvectors basis

, u₁, . . . , u_n orthonormal basis ?

u₁, . . . , un Rⁿ orthonormal basis , Q^TQ = In, inverse matrix 性, Q^T= Q⁻¹. Q column vectors Rⁿ orthonormal basis

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, Q⁻¹= Q^T. 性, n× n matrix column vectors

Rⁿ orthonormal basis , orthogonal matrix (

orthonormal matrix). A Q^TAQ = D, A orthogonal

diagonalizable.

Question 9.7. Q∈ Mn×n, Q⁻¹= Q^T Q orthogonal matrix?

, Q n× n orthogonal matrix D =



 λ1

. ..



 n× n diagonal

matrix Q^TAQ = D. AQ = QD, Q i-th column A eigenvalue λi

eigenvector, Q column vectors Rⁿ orthonormal basis, .

Proposition 9.4.2. A∈ Mn×n. A orthogonal diagonalizable n×n orthogonal matrix Q Q^TAQ diagonal matrix.

Proposition 9.4.2, A orthogonal diagonalizable Q, D∈ Mn×n

Q orthogonal matrix, D diagonal matrix A = QDQ^T. A^T= (QDQ^T)^T= (Q^T)^TD^TQ^T. (Q^T)^T= Q D^T= D ( D diagonal matrix), A^T= QDQ^T= A,

A symmetric. .

Corollary 9.4.3. A∈ Mn×n orthogonal diagonalizable, A symmetric matrix.

Spectral Theorem Corollary 9.4.3 .

A symmetric , A orthogonal diagonalizable. symmetric

matrix .

Lemma 9.4.4. A∈ Mn×n symmetric, v, w∈ Rⁿ ⟨Av,w⟩ = ⟨v,Aw⟩.

Proof. , , v, w∈ Rⁿ ⟨v,w⟩ = v^Tw

( v, w n× 1 matrix).

⟨Av,w⟩ = (Av)^Tw = (v^TA^T)w = v^T(A^Tw) =⟨v,A^Tw⟩.

A^T= A ⟨Av,w⟩ = ⟨v,Aw⟩.

n× n matrix diagonalizable characteristic

polynomial 數 . symmetric matrix

characteristic polynomial 數 .

Lemma 9.4.5. A∈ Mn×n symmetric, A characteristic polynomial pA(t) .

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Proof. λ = a + bı ( ı 數 ı²=−1) p_A(t) , a, b∈ R

b̸= 0. 數 , entry 數 .

數 . a + bı pA(t) , A− (a + bı)In

0. A− (a + bı)In A− (a − bı)In

(A− (a + bı)In)(A− (a − bı)In) = A²− 2aA + (a²+ b²)I_n.

a, b∈ R A 數 , A²− 2aA + (a²+ b²)In 數 . det(A− (a + bı)In) = 0,

det(A²− 2aA + (a²+ b²)I_n) = det(A− (a + bı)In) det(A− (a − bı)In) = 0.

A²− 2aA + (a²+ b²)I_n singular, v∈ Rⁿ v̸= 0 (A²− 2aA + (a²+ b²)In)v = A²v− 2aAv + (a²+ b²)v = O.

⟨A²v− 2aAv + (a²+ b²)v, v⟩ = ⟨A²v, v⟩ − 2a⟨Av,v⟩ + a²⟨v,v⟩ + b²⟨v,v⟩.

A symmetric, Lemma 9.4.4 ⟨A²v, v⟩ = ⟨A(Av),v⟩ = ⟨Av,Av⟩,

⟨Av − av,Av − av⟩ + b²⟨v,v⟩ = ⟨A²v, v⟩ − 2a⟨Av,v⟩ + a²⟨v,v⟩ + b²⟨v,v⟩,

∥Av − av∥²+ b²∥v∥²=⟨A²v− 2aAv + (a²+ b²)v, v⟩ = ⟨O,v⟩ = 0.

∥Av − av∥ ≥ 0, ∥v∥ > 0, b = 0. 初 b̸= 0 , pA(t) = 0

, .

symmetric matrix characteristic polynomial ,

symmetric matrix orthogonal diagonalizable. 數學 ,

2× 2 symmetric matrix orthogonal diagonalizable. (n− 1) × (n− 1) symmetric matrix orthogonal diagonalizable. A n× n symmetric matrix orthogonal diagonalizable. Lemma 9.4.5 數 λ A eigenvalue. u₁ A λ eigenvector ∥u1∥ = 1. Gram-Schmidt process, u₁ Rⁿ orthonormal basis u1, . . . , un. orthogonal matrix Q =



 u ₁ u₂ ··· un



, j = 1, . . . , n Au_j= c_{1 j}u₁+···+cn ju_n,

AQ = QC, C = [ci j]. Q orthogonal matrix, C = Q⁻¹AQ = Q^TAQ.

A symmetric C^T= Q^TAQ = C, C symmetric.

Au1=λu1, C 1-st column





 λ

0 ... 0





, C symmetric C 1-st row

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[λ 0 ··· 0]. C

C =







λ 0 ··· 0 0

... B 0





.

C symmetric, B (n− 1) × (n − 1) symmetric matrix. ,

B orthogonal diagonalizable, w₁, . . . , w_n−1 Rⁿ⁻¹ orthonomal basis B eigenvectors. R =





w ₁ w₂ ··· wn−1



, R (n− 1) × (n − 1) orthogonal matrix (n− 1) × (n − 1) digonal matrix D R^TBR = D.

P =







1 0 ··· 0 0... R 0





. ,

P^TCP =







λ 0 ··· 0

0

... R^TBR 0





=







λ 0 ··· 0 0

... D 0





.

P^TCP diagonal matrix, (QP)^TA(QP) = P^T(Q^TAQ)P = P^TCP diagonal matrix. Q, P orthogonal matrix, (QP)^T(QP) = P^T(Q^TQ)P = P^TP = In,

QP orthogonal matrix. Proposition 9.4.2, A orthogonal diagonalizable, Spectral Theorem.

Theorem 9.4.6 (Spectral Theorem). A n×n symmetric matrix, A orthogonal diagonalizable.

, n× n symmetric matrix A, orthogonal matrix Q

Q^TAQ diagonal matrix. , Theorem 9.4.6 , 數學

步步 Q . Gram-Schmidt process, .

Proposition, 步 .

Proposition 9.4.7. A n× n symmetric matrix. v, w∈ Rⁿ A eigenvectors eigenvalue 數, ⟨v,w⟩ = 0.

Proof. v, w eigenvalue λ,λ^′. Av =λv,Aw = λ^′w.

⟨Av,w⟩ = ⟨λv,w⟩ = λ⟨v,w⟩. ⟨v,Aw⟩ =λ^′⟨v,w⟩. Lemma 9.4.4

⟨Av,w⟩ = ⟨v,Aw⟩, (λ − λ^′)⟨v,w⟩ = 0. λ ̸= λ^′ ⟨v,w⟩ = 0.

A n× n symmetric matrix, A eigenvectors

Rⁿ orthonormal basis. A eigenvaluesλ1, . . . ,λk, eigenspace E_A(λ1), . . . , E_A(λk). E_A(λi) basis ,

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A diagonalizable Rⁿ basis. Proposition 9.4.7 , λi̸=λj

, EA(λi) EA(λj) . EA(λi) basis

. Gram-Schmidt process A eigenspace

E_A(λi) orthonormal basis. eigenspace basis

, A eigenvectors Rⁿ orthonormal basis.

.

Example 9.4.8. (1) symmetric matrix A =



 0 1 1 1 1 0 1 0 1



. A characteristic polynomial pA(t) =−(t + 1)(t − 1)(t − 2). A eigenvalues, −1,1,2.

A , Proposition 9.4.7 eigenvector .

−1,1,2 eigenvector

v₁=



−2 1 1



, v₂=



 0

−1 1



, v₃=



1 1 1



.

. i = 1, 2, 3 u_i=_∥v¹

i∥v_i,

u₁= 1

√6



−2 1 1



, u₂= 1

√2



 0

−1 1



, u₃= 1

√3



1 1 1





R³ orthonromal basis. A





−^√²₆ ^√¹₆ ^√¹₆ 0 −^√¹₂ ^√¹₂

√1 3







 0 1 1 1 1 0 1 0 1









−^√²₆ 0 ^√¹

1 3

√6 −^√¹₂ ^√¹₃

√1 6

√1 2

√1 3



 =



 −1 0 0

0 1 0

0 0 2



.

(2) symmetric matrix B =



 5 −4 −2

−4 5 −2

−2 −2 8



. B characteristic poly-

nomial p_B(t) =−t(t − 9)². B eigenvalues 0, 9. B , dim(E_B(0)) = 1, dim(E_B(9)) = 9. v₁=



2 2 1



 E_B(0) = N(B) basis, v₂=



−1 1 0



,v₃=



−1 0 2



 E_B(9) basis. Proposition 9.4.7 ⟨v1, v₂⟩ = ⟨v1, v₃⟩ = 0, . ⟨v2, v₃⟩ = 1 ̸= 0, Gram-Schmidt process v₂, v₃ E_B(9) orthogonal basis. w₂= v₂

w₃= v₃− Projw2v₃=



−1 0 2



 −1 2



−1 1 0



 = 1 2



−1

−1 4



.

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u₁=_∥v¹

1∥v₁, u₂=_∥w¹

2∥w₂, u₃=_∥w¹

3∥w₃ u₁= 1

√3



2 2 1



, u₂= 1

√2



−1 1 0



, u₃= 1 3√

2



−1

−1 4





R³ orthonromal basis. B





2 3

1

−^√¹₂ ^√¹₂ 03

−₃^√¹₂ −₃^√¹₂ ₃^√⁴₂







 5 −4 −2

−4 5 −2

−2 −2 8









2

3 −^√¹₂ −₃^√¹₂

2 3

√1

2 −₃^√¹₂

1

3 0 ⁴

3√ 2



 =



 0 0 0 0 9 0 0 0 9



.

9.5. Application: Conics and Quadric Surfaces

symmetric matrix orthogonal diagonalizable 性

線 ,

.

線 .

, quadratic form . n

數 quadratic form

∑

n i, j=1

a_{i j}x_ix_j

. x²+ 3xy− y², 3x²+ y²− z²+ 5xy + xz + 3yz 數數 quadratic form. x =



 x₁

... xn



, n 數 quadratic form

x^TAx , A n× n symmetric matrix. 數 quadratic form

ax²₁+ bx1x2+ cx²₂

ax²₁+ bx₁x₂+ cx²₂=[

x₁ x₂ ][ a b/2 b/2 c

][x₁ x₂ ]

. 數 quadratic form ax²₁+ bx²₂+ cx²₃+ rx1x2+ sx1x3+ tx2x3

ax²₁+ bx²₂+ cx²₃+ rx1x2+ sx1x3+ tx2x3=[

x1 x2 x3

]

 a r/2 s/2 r/2 b t/2 s/2 t/2 c







x1

x2

x3



.

quadratic form A symmetric, orthogonal

matrix Q Q^TAQ diagonal matrix



 λ1

. ..

λn



. 數 x =



 x₁

... xn





t =



 t1

... tn



 t = Q^Tx ( Q^T= Q⁻¹, x = Q t),

x^TAx = (Q t)^TA(Q t) = t^T(Q^TAQ) t =[

t1 ··· tn

]



 λ1

. ..

λn







 t₁

... t_n



 = λ¹t₁²+··· +λnt_n².

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9.5. Application: Conics and Quadric Surfaces 205

, 數 quadratic form .

.

Example 9.5.1. quadratic form x²₁+ 4x1x2− 2x²₂. x²₁+ 4x₁x2− 2x²₂=[

x1 x2

][ 1 2 2 −2

][x₁ x2

] . [ 1 2

2 −2 ]

symmetric matrix, orthogonal diagonalizable, [ 2/√

5 1/√ 5

−1/√

5 2/√ 5

][ 1 2 2 −2

][ 2/√

5 −1/√ 5 1/√

5 2/√ 5

]

=

[ 2 0 0 −3

] . [t1

t2

]

=

[ 2/√

5 1/√ 5

−1/√

5 2/√ 5

][x1

x2

]

[ x1 x2

][ 1 2 2 −2

][x1

x2

]

=[ t1 t2

][ 2 0 0 −3

][t1

t2

]

= 2t₁²− 3t₂². quadratic form x²₂+ x²₃+ 2x₁x₂+ 2x₁x₃,

x²₂+ x²₃+ 2x₁x₂+ 2x₁x₃=[

x1 x2 x3

]

 0 1 1 1 1 0 1 0 1







x1

x2

x₃



.

Example 9.4.8 Q^T



 0 1 1 1 1 0 1 0 1



Q =



 −1 0 0

0 1 0

0 0 2



 Q orthogonal

matrix





−^√²₆ 0 √¹ 1 3

√6 −^√¹₂ ^√¹₃

√1 6

√1 2

√1 3



.



t1

t2

t3



 =





−^√²₆ ^√¹₆ ^√¹₆ 0 −^√¹₂ ^√¹₂

√1 3







x1

x2

x3





[ x₁ x₂ x₃ ]

 0 1 1 1 1 0 1 0 1







x₁ x₂ x3



 = [ t₁ t₂ t₃ ]

 −1 0 0

0 1 0

0 0 2







t₁ t₂ t3



 = −t1²+ t₂²+ 2t₃².

線 . 線 ax²+ bxy +

cy²+ dx + ey + f = 0. ,

[ x y ][ a b/2 b/2 c

][x y ]

+[

d e ][x y ]

+ f = 0. (9.5)

symmetric matrix A =

[ a b/2 b/2 c

]

Q^TAQ =

[ λ1 0 0 λ2

] . 數

[x y ]

= Q^T [x

y ]

(

[x y ]

= Q [x

y ]

), (9.5)

[ x y ][ λ1 0 0 λ2

][x y ]

+[

d e ] Q

[x y ]

+ f = 0.

λ1x²+λ2y²+ d^′x + e^′y + f = 0, (9.6) [ d^′ e^′ ]

=[

d e ] Q.