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

Page 1

Review

⌅⌅ minimal solution of Ax = b :

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

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

Then

⌅ s 2 R(LA^⇤).

⌅ s is the only solution in R(L_A⇤).

That is, AA^⇤u = b ) s = A^⇤u.

⌅ s is unique.

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

K = {s} + KH.

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 f_T(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 ] .

Page 3

⌅ T is diagonalizable ) [T ] is diagonal for some . ) [T ^⇤] = [T ]^⇤ is diagonal.

) [T T ^⇤] = [T ] [T ]^⇤ = [T ]^⇤[T ] = [T^⇤T ] [diagonal]

) T T ^⇤ = T ^⇤T [rep is unique]

Does this commutativity imply diagonalizability?

⌅ normal operator T on an inner product space: T T^⇤ = T^⇤T

⌅ normal matrix A : AA^⇤ = A^⇤A

⌅ T (A) is normal , [T ] is normal.

N0

엔

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]

T.T_t.TT^*

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dy

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

Page 6

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

2. f_T(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 .

is

set

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]

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, v_ii = hvi, _iv_ii = hvi, U (v_i)i = 0, i = 1, · · · , n ) i = 0, i = 1, · · · , n

) 8x 2 V,

U (x) = U (P_n

i=1 a_iv_i) = P_n

i=1 a_iU (v_i) = P_n

i=1 a_{i u}v_i = 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^⇤ = T^⇤T = I ) hT (x), T (y)i = hx, T ^⇤T (y)i = hx, yi

unitary.arthogor.ae

Page 10

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

) hT (vi), T (v_j)i = hvi, v_ji = 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 (P_n

i=1 a_iv_i)||² = || P_n

i=1 a_iT (v_i)||²

= hP_n

i=1 a_iT (v_i), P_n

j=1 a_jT (v_j)i

= P_n

i=1

P_n

j=1 a_ia_jhT (vi), T (v_j)i = P_n

i=1 |ai|² = ||x||²

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

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

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

) T^⇤T = I ) T T^⇤ = I [Thm 2.17c]

윫

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 _✓ ) T✓T_✓^⇤ = T_✓T _✓ = I

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

⌅⌅ unitary matrix: AA^⇤ = A^⇤A = I

⌅ orthogonal matrix over the real field : AA^t = A^tA = 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 = Q^⇤BQ.

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

⌅ 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 [L_A] = A.

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

) [LA] = D, diagonal.

퓷

Page 13

) A = [LA] = [I] [L_A] [I] = Q ¹DQ.

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

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

) vj = P_n

i=1hvj, u_iiui ) Qij = hvj, u_ii ) Q^⇤_ij = Q_ji = hvi, u_ji = huj, v_ii = ([I] )ij

) QQ^⇤ = [I] [I] = I, and similarly, Q^⇤Q = I.

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

) AA^⇤ = Q^⇤DQ(Q^⇤DQ)^⇤ = Q^⇤DQQ^⇤D^⇤Q

= Q^⇤DD^⇤Q = Q^⇤D^⇤DQ = Q^⇤D^⇤QQ^⇤DQ = (Q^⇤DQ)^⇤Q^⇤DQ =

= A^⇤A [diagonals commute]

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 ); f_A(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.

⑧

their

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

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 = P_k

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 eis^{en Values)}

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, v_ii = hvi, _iv_ii = hvi, U (v_i)i = 0, i = 1, · · · , n ) i = 0, i = 1, · · · , n

) 8x 2 V,

U (x) = U (P_n

i=1 a_iv_i) = P_n

i=1 a_iU (v_i) = P_n

i=1 a_{i i}v_i = 0

T.^' Normal .TT#=TT

⇒ 丁以二 干叉T I

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, T^⇤T (y)i = hx, yi

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

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

)”4” : ||T (x)||² = ||T (P_n

i=1 a_iv_i)||² = hP_n

i=1 a_iT (v_i), P_n

j=1 a_jT (v_j)i

= P_n

i=1 P_n

j=1 a_ia_jhT (vi), T (v_j)i = P_n

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

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

Page 4

⌅⌅ unitary matrix: AA^⇤ = A^⇤A = I

⌅ orthogonal matrix over the real field : AA^t = A^tA = 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 = Q^⇤BQ(or Q^tBQ).

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

, A is unitarily equivalent to a diagonal matrix.

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

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

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

) A = Q^⇤DQ since Q = [I] is unitary for orthonormal and . (* Qij = hvj, u_ii, Q^⇤_ij = Q_ji = hvi, u_ji = huj, v_ii = ([I] )ij)

Page 5

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

, A = Q^tDQ, where D is a real diagonal matrix.

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

triangular matrix.

2. If F = R, then A = Q^tU 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 ).

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 = P_k

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

⌅ The orthogonal projection on W provides the best approximation.

⌅ [End of Review]

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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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

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 · · · W_k orthogonal projection on W_i T₁ · · · Tk

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

1. V = W₁ · · · W_k. 2. W_i^? = ^k_j=1,j6=iW_j. 3. T_iT_j = T₀, i 6= j.

4. T₁ + · · · + T_k = I.

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

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Page 10

proof of 5: x 2 V

) x = x1 + · · · + x_k, x_i 2 Wi [1]

T (x) = T (x₁) + · + T (xk)

= ₁x₁ + · · · + kx_k

= ₁T₁(x) + · · · + kT_k(x)

= ( ₁T₁ + · · · + kT_k)(x)

Page 11

⌅⌅ example: consider T = L_A, 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 = L_A1,

2 = 3, W₂ = {(0, a2, a₃, 0)^t : a₂, a₃ 2 R}, T2 = L_A2,

3 = 5, W₃ = {(0, 0, 0, a4)^t : a₄ 2 R}, T3 = L_A3, 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 : L_A = 2L_A1 + 3L_A2 + 5L_A3

Page 12

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

✓ 1 2 2 1

◆ .

1 = 3, W₁ = {(a1, a₂)^t 2 R² : a₁ = a₂}, T1 = L_A1,

2 = 1, W₂ = {(a1, a₂)^t 2 R² : a₁ + a₂ = 0}, T2 = L_A2, where A₁ = ¹₂

✓ 1 1 1 1

◆

and A₂ = ¹₂

✓ 1 1

1 1

◆ . T = ₁T₁ + ₂T₂ : L_A = 3L_A1 L_A2

(

E. 涎一

vi. 단

:

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 + · · · + _kT_k 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 + · · · + kT_k)( ₁T₁^⇤ + ₂T₂^⇤ + · · · + kT_k^⇤)

= | 1|²T₁T₁^⇤ + | 2|²T₂T₂^⇤ + · · · + | k|²T_kT_k^⇤

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

= I

Hence T is unitary.

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 + · · · + kT_k = ₁T₁ + ₂T₂ + · · · +

kT_k ) i is real.

”(” Let T = 1T₁ + ₂T₂ + · · · + _kT_k be the spectral decom- position of T . Suppose that every eigenvalue of T is real. Then T_i^⇤(v_i) = _iv_i = T_i(v_i), so T^⇤ = ₁T₁ + ₂T₂ + · · · + kT_k =

1T₁ + ₂T₂ + · · · + kT_k) = T.

e . 一一

t t ^*

Page 15

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

T =

 cos2✓ sin2✓

sin2✓ cos2✓

T ^{= T}

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