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.

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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 (x_{1}) = _{1}x_{1}; T (x_{2}) = _{2}x_{2}; _{1} 6= 2 ) hx1, x_{2}i = 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

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 .

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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, · · · , v^{n}} orthonormal and consisting of eigenvectors
of U. [Thm 6.16, 6.17]

) ihvi, v_{i}i = hvi, _{i}v_{i}i = 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_{i}v_{i}) = P_{n}

i=1 a_{i}U (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 R^{2} 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

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Page 10

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

) hT (vi), T (v_{j})i = hvi, v_{j}i = ij

) {T (vi), · · · , T (v^{n})} is an orthonormal basis.

”3)4” : Obvious.

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

) ||T (x)||^{2} = ||T (P_{n}

i=1 a_{i}v_{i})||^{2} = || P_{n}

i=1 a_{i}T (v_{i})||^{2}

= hP_{n}

i=1 a_{i}T (v_{i}), P_{n}

j=1 a_{j}T (v_{j})i

= P_{n}

i=1

P_{n}

j=1 a_{i}a_{j}hT (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 ) = T_{0}, [(I T^{⇤}T ) is self-adjoint; Lemma6.18]

) T^{⇤}T = 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 R^{2}

⌅ 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^{t}A = I

⌅ real unitary matrix = orthogonal matrix

⌅ columns from an orthonormal basis for F ^{n}.

⌅ rows form an orthonormal basis for F^{n}.

⌅ 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^{t}BQ.

⌅ 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 F^{n}.
) LA is normal and [L_{A}] = A.

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

) [LA] = D, diagonal.

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Page 13

) A = [LA] = [I] [L_{A}] [I] = Q ^{1}DQ.

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

Let = (u_{1}, · · · , un) and = (v_{1}, · · · , vn), both orthonormal.

) vj = P_{n}

i=1hvj, u_{i}iui ) Qij = hvj, u_{i}i
) Q^{⇤}_{ij} = Q_{ji} = hvi, u_{j}i = huj, v_{i}i = ([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.

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Page 15

Orthogonal projection and spectral theorem

⌅⌅ LetV = W_{1} L

W_{2} for subspaces W_{1} and W_{2}.

Then, 8x 2 V, x = x1 + x_{2} for some s_{1} 2 W1, x_{2} 2 W2.

⌅ projection T on W_{1} along W_{2} : T (x_{1} + x_{2}) = x_{1}

⌅ R(T ) = W_{1} = {x 2 V : T (x) = x}

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

⌅ For a projection T on W_{1},we can choose various W_{2}.

⌅ If T is an ”orthogonal” projection on W_{1}, then W_{2} is unique.

⌅⌅ T is a projection , T = T^{2} [alt def]

proof: ” ) ” : 8x 2 V, T^{2}(x) = T (x_{1}) = T (x_{1} + 0) = x_{1}

” ( ”: Assume that T = T ^{2}.

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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 ) T^{2}(u) = T (u) = x
x 2 N(T ) ) T (x) = 0 ) T^{2}(x) = T (x) = 0

) x = 0, (* T^{2}(u) = T (x) = x = 0)
Now we show that V = R(T ) + N(T ).

8x 2 V , let x1 = T (x) and x_{2} = x x_{1}. ) x1 2 R(T )

T (x_{2}) = T (x) T (x_{1}) = T (x) T^{2}(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 , T^{2} = T = T^{⇤}.
Proof:

” ) ”: Assume T is an orthogonal projection.

) T^{2} = T [projection]

) V = R(T ) L

N (T ); R(T )^{?} = N (T )

) 8x, y 2 V, x x1 + x_{2} and y = y_{1} + y_{2}, for some x_{1}, y_{1} 2 R(T )
and x_{2}, y_{2} 2 N(T ).

) hx, T (y)i = hx1 + x_{2}, y_{1}i = hx1, y_{1}i + hx2, y_{1}i = hx1, y_{1}i =
hx1, y_{1}i + hx1, y_{2}i = hx1, y_{1} + y_{2}i = hT (x), yi = hx, T^{⇤}(y)i

” ( ”: Assume T^{2} = 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 + x_{2}, x_{1} 2 R(T ), x2 2 N(T ) [projection]

) 0 = hx, x2i = hx1, x_{2}i + hx2, x_{2}i = ||x2||^{2} [x_{1} 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 + y_{2}, y_{1} 2 R(T ), y2 2 N(T ) [projection]

) 0 = hy, y1i = hy1, y_{1}i + hy2, y_{1}i = ||y1||^{2}[y_{2} 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 .

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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, · · · , v^{n}} orthonormal and consisting of eigenvectors
of U. [Thm 6.16, 6.17]

) ihvi, v_{i}i = hvi, _{i}v_{i}i = 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_{i}v_{i}) = P_{n}

i=1 a_{i}U (v_{i}) = P_{n}

i=1 a_{i i}v_{i} = 0

T.^{'} Normal .TT#=TT

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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, T^{⇤}T (y)i = hx, yi

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

) {T (vi), · · · , T (v^{n})} is an orthonormal basis.

)”4” : ||T (x)||^{2} = ||T (P_{n}

i=1 a_{i}v_{i})||^{2} = hP_{n}

i=1 a_{i}T (v_{i}), P_{n}

j=1 a_{j}T (v_{j})i

= P_{n}

i=1 P_{n}

j=1 a_{i}a_{j}hT (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^{t}A = 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^{t}BQ).

⌅⌅ 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 ^{1}DQ

) A = Q^{⇤}DQ since Q = [I] is unitary for orthonormal and .
(* Qij = hvj, u_{i}i, Q^{⇤}_{ij} = Q_{ji} = hvi, u_{j}i = huj, v_{i}i = ([I] )ij)

Page 5

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

, A = Q^{t}DQ, 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^{t}U Q, where U is a real upper triangular
matrix.

Orthogonal projection and spectral theorem

⌅ projection T on W_{1} along W_{2} : T (x_{1} + x_{2}) = x_{1}

⌅ R(T ) = W_{1} = {x 2 V : T (x) = x}

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

) V = R(T ) L

N (T ).

Page 6

⌅ For a projection T on W_{1},we can choose various W_{2}.

⌅ If T is an ”orthogonal” projection on W_{1}, then W_{2} is unique.

⌅⌅ T is a projection , T = T^{2} [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 , T^{2} = T = T^{⇤}.
Proof:

” ) ”: Assume T is an orthogonal projection.

) T^{2} = T [projection]

) V = R(T ) L

N (T ); R(T )^{?} = N (T )

) 8x, y 2 V, x = x1+ x_{2} and y = y_{1}+ y_{2}, for some x_{1}, y_{1} 2 R(T )
and x_{2}, y_{2} 2 N(T ).

) hx, T (y)i = hx1 + x_{2}, y_{1}i = hx1, y_{1}i + hx2, y_{1}i = hx1, y_{1}i =
hx1, y_{1}i + hx1, y_{2}i = hx1, y_{1} + y_{2}i = hT (x), yi = hx, T^{⇤}(y)i

” ( ”: Assume T^{2} = 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 + x_{2}, x_{1} 2 R(T ), x2 2 N(T ) [projection]

) 0 = hx, x2i = hx1, x_{2}i + hx2, x_{2}i = ||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 + y_{2}, y_{1} 2 R(T ), y2 2 N(T ) [projection]

) 0 = hy, y1i = hy1, y_{1}i + hy2, y_{1}i = ||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 W_{1} · · · W_{k}
orthogonal projection on W_{i} T_{1} · · · Tk

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

1. V = W_{1} · · · W_{k}.
2. W_{i}^{?} = ^{k}_{j=1,j}_{6=i}W_{j}.
3. T_{i}T_{j} = T_{0}, i 6= j.

4. T_{1} + · · · + T_{k} = I.

5. T = _{1}T_{1} + · · · + _{k}T_{k}: spectral decomposition
a

### ※썅i

Page 10

proof of 5: x 2 V

) x = x1 + · · · + x_{k}, x_{i} 2 Wi [1]

T (x) = T (x_{1}) + · + T (xk)

= _{1}x_{1} + · · · + kx_{k}

= _{1}T_{1}(x) + · · · + kT_{k}(x)

= ( _{1}T_{1} + · · · + 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, W_{1} = {(a1, 0, 0, 0)^{t} : a_{1} 2 R}, T1 = L_{A}_{1},

2 = 3, W_{2} = {(0, a2, a_{3}, 0)^{t} : a_{2}, a_{3} 2 R}, T2 = L_{A}_{2},

3 = 5, W_{3} = {(0, 0, 0, a4)^{t} : a_{4} 2 R}, T3 = L_{A}_{3},
where A_{1} =

0 BB

@

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

1 CC

A , A^{2} =
0
BB

@

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

1 CC

A, and A^{3} =
0
BB

@

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

1 CC A.

T = _{1}T_{1} + _{2}T_{2} + _{3}T_{3} : L_{A} = 2L_{A}_{1} + 3L_{A}_{2} + 5L_{A}_{3}

Page 12

⌅⌅ example: consider T = L_{A}, where A =

✓ 1 2 2 1

◆ .

1 = 3, W_{1} = {(a1, a_{2})^{t} 2 R^{2} : a_{1} = a_{2}}, T1 = L_{A}_{1},

2 = 1, W_{2} = {(a1, a_{2})^{t} 2 R^{2} : a_{1} + a_{2} = 0}, T2 = L_{A}_{2},
where A_{1} = ^{1}_{2}

✓ 1 1 1 1

◆

and A_{2} = ^{1}_{2}

✓ 1 1

1 1

◆
.
T = _{1}T_{1} + _{2}T_{2} : L_{A} = 3L_{A}_{1} L_{A}_{2}

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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 = 1T_{1} + _{2}T_{2} + · · · + _{k}T_{k} be the spectral decompo-
sition of T . If | | = 1 for every eigenvalue of T , then by 3. of the
spectral theorem,

T T^{⇤} = ( _{1}T_{1} + _{2}T_{2} + · · · + kT_{k})( _{1}T_{1}^{⇤} + _{2}T_{2}^{⇤} + · · · + kT_{k}^{⇤})

= | 1|^{2}T_{1}T_{1}^{⇤} + | 2|^{2}T_{2}T_{2}^{⇤} + · · · + | k|^{2}T_{k}T_{k}^{⇤}

= (T_{1} + T_{2} + · · · + Tk)(T_{1} + T_{2} + · · · + 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 ) 1T_{1} + _{2}T_{2} + · · · + kT_{k} = _{1}T_{1} + _{2}T_{2} + · · · +

kT_{k} ) i is real.

”(” Let T = 1T_{1} + _{2}T_{2} + · · · + _{k}T_{k} be the spectral decom-
position of T . Suppose that every eigenvalue of T is real. Then
T_{i}^{⇤}(v_{i}) = _{i}v_{i} = T_{i}(v_{i}), so T^{⇤} = _{1}T_{1} + _{2}T_{2} + · · · + kT_{k} =

1T_{1} + _{2}T_{2} + · · · + 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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