K = { s } + K . ⌅⌅ x = s + y,s 2 R ( L ) ,y 2 N ( L ) ⌅ s isunique. Thatis, AA u = b ) s = A u. ⌅ s istheonlysolutionin R ( L ) . ⌅ s 2 R ( L ) . Then tionof Ax = b. ⌅⌅ Theorem6.13: A 2 M ( F ); b 2 F ; and s isaminimalsolu- solutionwiththesmallestnorm:le

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Page 1

Review

⌅⌅ minimal solution of Ax = b :

solution with the smallest norm:least energy, least power, etc.

⌅⌅ Theorem 6.13: A 2 Mm⇥n(F ); b 2 Fm; and s is a minimal solu- tion of Ax = b.

Then

s 2 R(LA).

s is the only solution in R(LA).

That is, AAu = b ) s = Au.

s is unique.

⌅⌅ Compare x = s + y, s 2 R(LA), y 2 N(LA) vs Theorem 3.9:

K = {s} + KH.

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Page 2

Normal and self-adjoint operator

⌅⌅ We now investigate diagonalizability of a linear operator on an inner product space.

⌅⌅ Lemma 6.14 V is an inner product space; dim(V ) < 1; T : V ! V is a linear operator. Then

T has an eigenvector ) T has an eigenvector.

⌅⌅ Theorem 6.14 (Schur): din(V ) < 1; T : V ! V is a linear opera- tor; and fT(t) splits. Then 9 an orthonormal basis such that [T ] is upper triangular.

Schur’s theorem does not say that [T ] is invertible.

Neither does it say that [T ] is diagonalizable.

in the theorem is not unique; nor [T ] .

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Page 3

T is diagonalizable ) [T ] is diagonal for some . ) [T ] = [T ] is diagonal.

) [T T ] = [T ] [T ] = [T ][T ] = [TT ] [diagonal]

) T T = T T [rep is unique]

Does this commutativity imply diagonalizability?

normal operator T on an inner product space: T T = TT

normal matrix A : AA = AA

T (A) is normal , [T ] is normal.

N0

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Page 4

⌅⌅ Theorem 6.15 : V is an inner product space; T : V ! V is normal.

Then

1. 8x 2 V, ||T (x)|| = ||T(x)||.

2. 8c 2 F, T cI is normal.

3. T (x) = x , T(x) = x. That is, T and T have the same eigenvector x with the respective eigenvalues and .

4. T (x1) = 1x1; T (x2) = 2x2; 1 6= 2 ) hx1, x2i = 0.

⌅⌅ Theorem 6.16 : V is an inner product space over the ”complex”

field; dim(V ) < 1; T : V ! V is linear. Then T is normal if and only if there exists an orthonormal basis for V consisting of eigenvectors of T .

⌅⌅ [End of Review]

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Page 5

⌅⌅ self-adjoint (Hermitian) operator T on an inner product space:

T = T .

self-adjoint (Hermitian) matrix A: A = A

these definitions apply to both the real and complex field.

For complex matrices, self-adjoint means conjugate symmetric.

For real matrices, self-adjoint means symmetric.

For orthonormal , T = T , [T ] = [T ]

self-adjoint ) normal; normal 6) self-adjoint

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⌅⌅ Lemma 6.17: V is an inner product space; dim(V ) < 1; T : 1. Every eigenvalue of T is real.

2. fT(t) splits over the real field.

proof: ”1”: Let T (x) = x, x 6= 0.

) hx, xi = hT (x), xi = hx, T(x)i = hx, T (x)i= hx, xi = hx, xi

2 follows from 1.

⌅⌅ Theorem 6.17: V is an inner product space over the ”real” field;

dim(V ) < 1; T : V ! V is linear. Then T is self-adjoint if and only if there exists an orthonormal basis for V consisting of eigenvectors of T .

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Page 7

proof:

”only if”: Assume T = T.

) fT(t) splits (over the real field). [Lemma 6.17]

) 9 , orthonormal, such that [T ] is upper triangular. [Schur’s Thm]

) [T ] = [T] = [T ] [self-adjoint]

) [T ] is (real) diagonal.

) consists of eigenvectors of T .

”if” : Assume an orthonormal basis of eigenvectors of T . ) [T ] is (real) diagonal.

) [T ] = [T ] = [T ]

) T = T [rep is unique]

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Page 8

Unitary and orthogonal operator

⌅⌅ V is an inner product space; dim(V ) < 1; T : V ! V is linear.

unitary operator: 8x 2 V (C), ||T (x)|| = ||x||

orthogonal operator: 8x 2 V (R), ||T (x)|| = ||x||

⌅⌅ Lemma 6.18: V is an inner product space; dim(V ) < 1; U : V ! V is self-adjoint. Then 8x 2 V, hx, U(x)i = 0 ) U = T0.

proof.

Assume U is self-adjoint and 8x 2 V, hx, U(x)i = 0.

) 9 = {v1, · · · , vn} orthonormal and consisting of eigenvectors of U. [Thm 6.16, 6.17]

) ihvi, vii = hvi, ivii = hvi, U (vi)i = 0, i = 1, · · · , n ) i = 0, i = 1, · · · , n

) 8x 2 V,

U (x) = U (Pn

i=1 aivi) = Pn

i=1 aiU (vi) = Pn

i=1 ai uvi = 0

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Page 9

⌅⌅ Theorem 6.18: V is an inner product space; dim(V ) < 1; T : V ! V is linear. Then these are all equivalent.

1. T T = T T = I

2. 8x, y 2 V, hT (x), T (y)i = hx, yi

3. is an orthonormal basis ) T ( ) is an orthonormal basis.

4. There exists an orthonormal basis such that T ( ) is an or- thonormal basis.

5. 8x 2 V, ||T (x)|| = ||x||

5 means length-preserving; 2 means sort of angle-preserving.

rotation and reflection in R2 are orthogonal.

unitary or orthogonal ) normal; not conversely.

proof: ”1)2”: Assume T T = TT = I ) hT (x), T (y)i = hx, T T (y)i = hx, yi

unitary.arthogor.ae

.

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Page 10

”2)3” : Assume ”2” and an orthonormal basis {v1, · · · , vn}.

) hT (vi), T (vj)i = hvi, vji = ij

) {T (vi), · · · , T (vn)} is an orthonormal basis.

”3)4” : Obvious.

”4)5” : Assume = {v1, · · · , vn} and T ( ) are orthonormal.

) ||T (x)||2 = ||T (Pn

i=1 aivi)||2 = || Pn

i=1 aiT (vi)||2

= hPn

i=1 aiT (vi), Pn

j=1 ajT (vj)i

= Pn

i=1

Pn

j=1 aiajhT (vi), T (vj)i = Pn

i=1 |ai|2 = ||x||2

”5)1”: Assume 8x 2 V, ||T (x)|| = ||x||.

) hx, xi = hT (x), T (x)i = hx, TT (x)i ) 8x 2 V, hx, (I T T )(x)i = 0

) (I T T ) = T0, [(I TT ) is self-adjoint; Lemma6.18]

) TT = I ) T T = I [Thm 2.17c]

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Page 11

⌅⌅ T is unitary or orthogonal; v is an eigenvector.

) ||v|| = ||T (v)|| = || v|| = | |||v||

) | | = 1

⌅⌅ example: linear operators on R2

T: rotation by ✓; T = T ) TT = TT = I

T: reflection about a line or the origin; T = T! T T = T T = I

⌅⌅ unitary matrix: AA = AA = I

orthogonal matrix over the real field : AAt = AtA = I

real unitary matrix = orthogonal matrix

columns from an orthonormal basis for F n.

rows form an orthonormal basis for Fn.

T is unitary , [T ] is unitary for an orthonormal .

T is orthogonal , [T ] is orthogonal for an orthonormal .

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Page 12

⌅⌅ A matrix A is unitarily equivalent to B : 9 a unitary matrix Q such that A = QBQ.

A real matrix A is textbforthogonally equivalent to real B : 9 an orthogonal matrix Q such that A = QtBQ.

A and B are unitarily equivalent ) they are similar, but not con- versely.

⌅⌅ Theorem 6.19: A complex n ⇥ n matrix A is normal.

, A is unitarily equivalent to a diagonal matrix.

This is the matrix version of Theorem 6.16.

proof: ”)”: Assume A is normal and is the std basis for Fn. ) LA is normal and [LA] = A.

) 9 an orthonormal basis of eigenvectors of LA. [Thm 6.16]

) [LA] = D, diagonal.

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Page 13

) A = [LA] = [I] [LA] [I] = Q 1DQ.

We show that Q = [I] is unitary if and are orthonormal.

Let = (u1, · · · , un) and = (v1, · · · , vn), both orthonormal.

) vj = Pn

i=1hvj, uiiui ) Qij = hvj, uii ) Qij = Qji = hvi, uji = huj, vii = ([I] )ij

) QQ = [I] [I] = I, and similarly, QQ = I.

”(”: Assume Q, unitary, and D, diagonal, are such that A = QDQ.

) AA = QDQ(QDQ) = QDQQDQ

= QDDQ = QDDQ = QDQQDQ = (QDQ)QDQ =

= AA [diagonals commute]

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Page 14

⌅⌅ Theorem 6.20: A real n ⇥ n matrix A is symmetric.

, A is orthogonally equivalent to a real diagonal matrix.

This is the matrix version of Theorem 6.17.

⌅⌅ Theorem 6.21 (Schur) : A 2 Mn⇥n(F ); fA(t) splits over F . Then

1. If F = C, then A is unitarily equivalent to a complex upper triangular matrix.

2. If F = R, then A is orthogonally equivalent to a real upper triangular matrix.

This is the matrix version of Schur’s Theorem 6.14.

their

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Page 15

Orthogonal projection and spectral theorem

⌅⌅ LetV = W1 L

W2 for subspaces W1 and W2.

Then, 8x 2 V, x = x1 + x2 for some s1 2 W1, x2 2 W2.

projection T on W1 along W2 : T (x1 + x2) = x1

R(T ) = W1 = {x 2 V : T (x) = x}

N (T ) = W2 = {x 2 V : T (x) = 0}

For a projection T on W1,we can choose various W2.

If T is an ”orthogonal” projection on W1, then W2 is unique.

⌅⌅ T is a projection , T = T2 [alt def]

proof: ” ) ” : 8x 2 V, T2(x) = T (x1) = T (x1 + 0) = x1

” ( ”: Assume that T = T 2.

K

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Page 16

First we show that R(T ) \ N(T ) = {0}.

Assume x 2 R(T ) \ N(T ).

x 2 R(T ) ) 9u such that T (u) = x ) T2(u) = T (u) = x x 2 N(T ) ) T (x) = 0 ) T2(x) = T (x) = 0

) x = 0, (* T2(u) = T (x) = x = 0) Now we show that V = R(T ) + N(T ).

8x 2 V , let x1 = T (x) and x2 = x x1. ) x1 2 R(T )

T (x2) = T (x) T (x1) = T (x) T2(x) = 0 ) x2 2 N(T ) V = R(T ) L

N (T )

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Page 17

⌅⌅ orthogonal projection T :

R(T )? = N (T ) and N(T )? = R(T )

If dim(V ) < 1, (R(T )? = N (T ) , N(T )? = R(T )).

Given a subspace W, T (y) = u, [Thm 6.6]

where y = u + z, u 2 W, z 2 W?,

defines an orthogonal projection on W .

A truncated Fourier series, for k < n, u = Pk

i=1hy, vii is an orthogonal projection on span({v1, · · · , vk}).

There is only one orthogonal projection on W .

The orthogonal projection on W provides the best approximation.

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Page 18

⌅⌅ Theorem 6.24: V is an inner product space:T : V ! V is linear.

Then T is an orthogonal projection , T2 = T = T. Proof:

” ) ”: Assume T is an orthogonal projection.

) T2 = T [projection]

) V = R(T ) L

N (T ); R(T )? = N (T )

) 8x, y 2 V, x x1 + x2 and y = y1 + y2, for some x1, y1 2 R(T ) and x2, y2 2 N(T ).

) hx, T (y)i = hx1 + x2, y1i = hx1, y1i + hx2, y1i = hx1, y1i = hx1, y1i + hx1, y2i = hx1, y1 + y2i = hT (x), yi = hx, T(y)i

” ( ”: Assume T2 = T = T ) T is a projection.

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Page 19

Let x 2 R(T ) and y 2 N(T ). ) T (x) = x, T (y) = 0.

) hx, yi = hT (x), yi = hx, T(y)i = hx, T (y)i = hx, 0i = 0 ) x 2 N(T )? and y 2 R(T )?

) R(T ) ✓ N(T )? and N(T ) ✓ R(T )? (1) Let x 2 N(T )?

) x = x1 + x2, x1 2 R(T ), x2 2 N(T ) [projection]

) 0 = hx, x2i = hx1, x2i + hx2, x2i = ||x2||2 [x1 2 N(T )? (1)]

) x2 = 0 ) x = x1 2 R(T ) ) N(T )? ✓ R(T ) ) N(T )? = R(T ) [(1), cf textbook]

Let y 2 R(T )?.

) y = y1 + y2, y1 2 R(T ), y2 2 N(T ) [projection]

) 0 = hy, y1i = hy1, y1i + hy2, y1i = ||y1||2[y2 2 R(T )? (1)]

) y1 = 0 ) y = y2 2 N(T ) ) R(T )? ✓ N(T ) ) R(T )? = N (T ) [(1), cf textbook]

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Page 1

Review

⌅⌅ self-adjoint (Hermitian) operator T : T = T .

self-adjoint (Hermitian) matrix A: A = A

A: Conjugate symmetric.

⌅⌅ Theorem 6.17: For finite V over the ”real” field, T is self-adjoint iff 9 an orthonormal basis for V consisting of eigenvectors of T . proof:

”only if”: Assume T = T.

) fT(t) splits (over the real field). [Lemma 6.17]

) 9 , orthonormal, s. t. [T ] is upper triangular. [Schur’s Thm]

) [T ] = [T] = [T ] [self-adjoint]

) [T ] is (real) diagonal.

) consists of eigenvectors of T .

(real eisen Values)

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Page 2

Unitary and orthogonal operator

⌅⌅ V is an inner product space; dim(V ) < 1; T : V ! V is linear.

unitary operator: 8x 2 V (C), ||T (x)|| = ||x||

orthogonal operator: 8x 2 V (R), ||T (x)|| = ||x||

⌅⌅ Lemma 6.18: V is an inner product space; dim(V ) < 1; U : V ! V is self-adjoint. Then 8x 2 V, hx, U(x)i = 0 ) U = T0.

proof.

Assume U is self-adjoint and 8x 2 V, hx, U(x)i = 0.

) 9 = {v1, · · · , vn} orthonormal and consisting of eigenvectors of U. [Thm 6.16, 6.17]

) ihvi, vii = hvi, ivii = hvi, U (vi)i = 0, i = 1, · · · , n ) i = 0, i = 1, · · · , n

) 8x 2 V,

U (x) = U (Pn

i=1 aivi) = Pn

i=1 aiU (vi) = Pn

i=1 ai ivi = 0

T.' Normal .TT#=TT

丁以二 干叉T I

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Page 3

⌅⌅ Theorem 6.18: The followings are all equivalent.

1. T T = T T = I

2. 8x, y 2 V, hT (x), T (y)i = hx, yi

3. 9 an orthonormal basis s. t. T ( ) is an orthonormal basis.

4. 8x 2 V, ||T (x)|| = ||x||

proof: ”1”: T T = T T = I

) ”2”: hT (x), T (y)i = hx, TT (y)i = hx, yi

)”3” for = {v1, · · · , vn}, hT (vi), T (vj)i = hvi, vji = ij

) {T (vi), · · · , T (vn)} is an orthonormal basis.

)”4” : ||T (x)||2 = ||T (Pn

i=1 aivi)||2 = hPn

i=1 aiT (vi), Pn

j=1 ajT (vj)i

= Pn

i=1 Pn

j=1 aiajhT (vi), T (vj)i = Pn

i=1 |ai|2 = ||x||2 )”1”: Assume 8x 2 V, ||T (x)|| = ||x||.

) hx, xi = hT (x), T (x)i = hx, TT (x)i ) TT = I ) T T = I [Thm 2.17c]

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Page 4

⌅⌅ unitary matrix: AA = AA = I

orthogonal matrix over the real field : AAt = AtA = I

T is unitary , [T ] is unitary for an orthonormal .

T is orthogonal , [T ] is orthogonal for an orthonormal .

⌅⌅ A matrix A is unitarily (or orthogonally) equivalent to B : 9 a unitary matrix Q such that A = QBQ(or QtBQ).

⌅⌅ Theorem 6.19: A is a complex and normal matrix.

, A is unitarily equivalent to a diagonal matrix.

proof: ”)”: LA is normal, where [LA] = A.

) 9 an orthonormal basis of eigenvectors of LA. [Thm 6.16]

) [LA] = D ) A = [LA] = [I] [LA] [I] = Q 1DQ

) A = QDQ since Q = [I] is unitary for orthonormal and . (* Qij = hvj, uii, Qij = Qji = hvi, uji = huj, vii = ([I] )ij)

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Page 5

⌅⌅ Theorem 6.20: A real n ⇥ n matrix A is symmetric(self-adjoint).

, A = QtDQ, where D is a real diagonal matrix.

⌅⌅ Theorem 6.21 (Schur) : A 2 Mn⇥n(F ); fA(t) splits over F . Then 1. If F = C, then A = QU Q, where U is a complex upper

triangular matrix.

2. If F = R, then A = QtU Q, where U is a real upper triangular matrix.

Orthogonal projection and spectral theorem

projection T on W1 along W2 : T (x1 + x2) = x1

R(T ) = W1 = {x 2 V : T (x) = x}

N (T ) = W2 = {x 2 V : T (x) = 0}

) V = R(T ) L

N (T ).

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Page 6

For a projection T on W1,we can choose various W2.

If T is an ”orthogonal” projection on W1, then W2 is unique.

⌅⌅ T is a projection , T = T2 [alt def]

⌅⌅ orthogonal projection T :

R(T )? = N (T ) and N(T )? = R(T )

If dim(V ) < 1, (R(T )? = N (T ) , N(T )? = R(T )).

Given a subspace W, T (y) = u, where y = u + z, u 2 W, z 2 W?, is an orthogonal projection on W .

A truncated Fourier series, for k < n, u = Pk

i=1hy, vii is an orthogonal projection on span({v1, · · · , vk}).

The orthogonal projection on W provides the best approximation.

[End of Review]

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Page 7

⌅⌅ Theorem 6.24: V is an inner product space:T : V ! V is linear.

Then T is an orthogonal projection , T2 = T = T. Proof:

” ) ”: Assume T is an orthogonal projection.

) T2 = T [projection]

) V = R(T ) L

N (T ); R(T )? = N (T )

) 8x, y 2 V, x = x1+ x2 and y = y1+ y2, for some x1, y1 2 R(T ) and x2, y2 2 N(T ).

) hx, T (y)i = hx1 + x2, y1i = hx1, y1i + hx2, y1i = hx1, y1i = hx1, y1i + hx1, y2i = hx1, y1 + y2i = hT (x), yi = hx, T(y)i

” ( ”: Assume T2 = T = T ) T is a projection.

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(27)

Page 8

Let x 2 R(T ) and y 2 N(T ). ) T (x) = x, T (y) = 0.

) hx, yi = hT (x), yi = hx, T(y)i = hx, T (y)i = hx, 0i = 0 ) x 2 N(T )? and y 2 R(T )?

) R(T ) ✓ N(T )? and N(T ) ✓ R(T )? (1) Let x 2 N(T )?

) x = x1 + x2, x1 2 R(T ), x2 2 N(T ) [projection]

) 0 = hx, x2i = hx1, x2i + hx2, x2i = ||x2||2, [*x1 2 N(T )? (1)]

) x2 = 0 ) x = x1 2 R(T ) ) N(T )? ✓ R(T ) ) N(T )? = R(T ) [(1)]

Let y 2 R(T )?.

) y = y1 + y2, y1 2 R(T ), y2 2 N(T ) [projection]

) 0 = hy, y1i = hy1, y1i + hy2, y1i = ||y1||2, [* y2 2 R(T )? (1)]

) y1 = 0 ) y = y2 2 N(T ) ) R(T )? ✓ N(T ) ) R(T )? = N (T ) [(1)]

方器

(28)

Page 9

⌅⌅ Theorem 6.25 (The spectral theorem): V is an inner product space over F ; dim(V ) < 1; T : V ! V is a linear operator with

distinct eigenvalues: spectrum 1 · · · k

corresponding eigenspaces W1 · · · Wk orthogonal projection on Wi T1 · · · Tk

and T is normal if F = C and self-adjoint if F = R. Then the following statements are true.

1. V = W1 · · · Wk. 2. Wi? = kj=1,j6=iWj. 3. TiTj = T0, i 6= j.

4. T1 + · · · + Tk = I.

5. T = 1T1 + · · · + kTk: spectral decomposition a

※썅i

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Page 10

proof of 5: x 2 V

) x = x1 + · · · + xk, xi 2 Wi [1]

T (x) = T (x1) + · + T (xk)

= 1x1 + · · · + kxk

= 1T1(x) + · · · + kTk(x)

= ( 1T1 + · · · + kTk)(x)

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Page 11

⌅⌅ example: consider T = LA, where A =

0 BB

@

2 0 0 0 0 3 0 0 0 0 3 0 0 0 0 5

1 CC A.

1 = 2, W1 = {(a1, 0, 0, 0)t : a1 2 R}, T1 = LA1,

2 = 3, W2 = {(0, a2, a3, 0)t : a2, a3 2 R}, T2 = LA2,

3 = 5, W3 = {(0, 0, 0, a4)t : a4 2 R}, T3 = LA3, where A1 =

0 BB

@

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

1 CC

A , A2 = 0 BB

@

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

1 CC

A, and A3 = 0 BB

@

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

1 CC A.

T = 1T1 + 2T2 + 3T3 : LA = 2LA1 + 3LA2 + 5LA3

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Page 12

⌅⌅ example: consider T = LA, where A =

✓ 1 2 2 1

◆ .

1 = 3, W1 = {(a1, a2)t 2 R2 : a1 = a2}, T1 = LA1,

2 = 1, W2 = {(a1, a2)t 2 R2 : a1 + a2 = 0}, T2 = LA2, where A1 = 12

✓ 1 1 1 1

and A2 = 12

✓ 1 1

1 1

◆ . T = 1T1 + 2T2 : LA = 3LA1 LA2

(

Th I - A ) o

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vi.

:

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Page 13

⌅⌅ Corollary 6.25.2: V is an inner product space over C; dim(V ) <

1; T : V ! V is unitary , T is normal and | | = 1 for every eigenvalue of T .

Proof.

”)” If T is unitary, then T is normal and every eigenvalue of T has absolute value 1 (* ||T(x)|| = ||x||).

”(” Let T = 1T1 + 2T2 + · · · + kTk be the spectral decompo- sition of T . If | | = 1 for every eigenvalue of T , then by 3. of the spectral theorem,

T T = ( 1T1 + 2T2 + · · · + kTk)( 1T1 + 2T2 + · · · + kTk)

= | 1|2T1T1 + | 2|2T2T2 + · · · + | k|2TkTk

= (T1 + T2 + · · · + Tk)(T1 + T2 + · · · + Tk)

= I

Hence T is unitary.

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Page 14

⌅⌅ Corollary 6.25.3: V is an inner product space over C; dim(V ) <

1; T : V ! V is normal. Then T is self-adjoint , every eigen- value of T is real.

Proof.

”)” T = T ) 1T1 + 2T2 + · · · + kTk = 1T1 + 2T2 + · · · +

kTk ) i is real.

”(” Let T = 1T1 + 2T2 + · · · + kTk be the spectral decom- position of T . Suppose that every eigenvalue of T is real. Then Ti(vi) = ivi = Ti(vi), so T = 1T1 + 2T2 + · · · + kTk =

1T1 + 2T2 + · · · + kTk) = T.

e . 一一

t t *

(34)

Page 15

⌅⌅ Reflection is an example that is both self adjoint and unitary.

T =

 cos2✓ sin2✓

sin2✓ cos2✓

T = T

( f

#T = I = ㅜㅜ

丁杆

愼坦剡

一二

T-T

£6 9 ) 導十

T-T II의 Here

T叫二

俊川汭

(

TT반

Ri )

Figure

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