Math 504 Fall 2016 NotesWeek 1, Part I

Emre Mengi

Department of MathematicsKoç University

Istanbul, Turkey

Emre Mengi Week 1, Lecture 1

Outline

Vector Norms

Inner or Dot Products

Orthogonality

Emre Mengi Week 1, Lecture 1

Vector Norms

Emre Mengi Week 1, Lecture 1

Commonly Used Vector Norms

Let v ∈ Cn.Euclidean length or vector 2-norm

‖v‖2 :=√|v1|2 + |v2|2 + · · ·+ |vn|2

1-norm ‖v‖1 := |v1|+ |v2|+ · · ·+ |vn|

∞-norm ‖v‖∞ := max{|v1|, |v2|, . . . , |vn|}

p-norm (p ∈ R, p ≥ 1)

‖v‖2 := p√|v1|p + |v2|p + · · ·+ |vn|p

Emre Mengi Week 1, Lecture 1

Commonly Used Vector Norms

Let v ∈ Cn.Euclidean length or vector 2-norm

‖v‖2 :=√|v1|2 + |v2|2 + · · ·+ |vn|2

1-norm ‖v‖1 := |v1|+ |v2|+ · · ·+ |vn|

∞-norm ‖v‖∞ := max{|v1|, |v2|, . . . , |vn|}

p-norm (p ∈ R, p ≥ 1)

‖v‖2 := p√|v1|p + |v2|p + · · ·+ |vn|p

Emre Mengi Week 1, Lecture 1

Commonly Used Vector Norms

Let v ∈ Cn.Euclidean length or vector 2-norm

‖v‖2 :=√|v1|2 + |v2|2 + · · ·+ |vn|2

1-norm ‖v‖1 := |v1|+ |v2|+ · · ·+ |vn|

∞-norm ‖v‖∞ := max{|v1|, |v2|, . . . , |vn|}

p-norm (p ∈ R, p ≥ 1)

‖v‖2 := p√|v1|p + |v2|p + · · ·+ |vn|p

Emre Mengi Week 1, Lecture 1

Commonly Used Vector Norms

Let v ∈ Cn.Euclidean length or vector 2-norm

‖v‖2 :=√|v1|2 + |v2|2 + · · ·+ |vn|2

1-norm ‖v‖1 := |v1|+ |v2|+ · · ·+ |vn|

∞-norm ‖v‖∞ := max{|v1|, |v2|, . . . , |vn|}

p-norm (p ∈ R, p ≥ 1)

‖v‖2 := p√|v1|p + |v2|p + · · ·+ |vn|p

Emre Mengi Week 1, Lecture 1

Definition

Definition (Norm)

A function ‖ · ‖ : Cn → R is called a norm if itsatisfies the following.

(i) ‖v‖ > 0 for all nonzero v ∈ Cn

(ii) ‖αv‖ = |α|‖v‖ for all α ∈ C and all v ∈ Cn

(iii) ‖v + u‖ ≤ ‖v‖+ ‖u‖ for all v ,u ∈ Cn

Emre Mengi Week 1, Lecture 1

Equivalence of Norms

TheoremAny two norms ‖ · ‖A and ‖ · ‖B are equivalent inthe following sense.

There exist positive constants C1,C2 ∈ R suchthat

C1‖v‖B ≤ ‖v‖A ≤ C2‖v‖B ∀v ∈ Cn.

For instance‖v‖∞ ≤ ‖v‖2 ≤

√n‖v‖∞.

(In this case, C1 = 1 and C2 =√

n.)

Emre Mengi Week 1, Lecture 1

Equivalence of Norms

TheoremAny two norms ‖ · ‖A and ‖ · ‖B are equivalent inthe following sense.

There exist positive constants C1,C2 ∈ R suchthat

C1‖v‖B ≤ ‖v‖A ≤ C2‖v‖B ∀v ∈ Cn.

For instance‖v‖∞ ≤ ‖v‖2 ≤

√n‖v‖∞.

(In this case, C1 = 1 and C2 =√

n.)

Emre Mengi Week 1, Lecture 1

Inner or Dot Products

Emre Mengi Week 1, Lecture 1

The Standard Inner Product

〈u, v〉S := u∗v =[

u1 u2 . . . un]

v1v2...

vn

= u1v1 + u2v2 + · · ·+ unvn

for vectors u, v ∈ Cn.

The 2-norm is induced by this standard inner product, that is

‖v‖2 =√|v1|2 + |v2|2 + · · ·+ |vn|2

=[

v1 v2 . . . vn]

v1v2...

vn

=√

v∗v =√〈v , v〉S

Emre Mengi Week 1, Lecture 1

The Standard Inner Product

〈u, v〉S := u∗v =[

u1 u2 . . . un]

v1v2...

vn

= u1v1 + u2v2 + · · ·+ unvn

for vectors u, v ∈ Cn.

The 2-norm is induced by this standard inner product, that is

‖v‖2 =√|v1|2 + |v2|2 + · · ·+ |vn|2

=[

v1 v2 . . . vn]

v1v2...

vn

=√

v∗v =√〈v , v〉S

Emre Mengi Week 1, Lecture 1

General Inner ProductsDefinition (Inner Product)

A function 〈·〉 : Cn × Cn → C is called an innerproduct if it satisfies the following.

(i) 〈v , v〉 > 0 for all nonzero v ∈ Cn

(ii) 〈u, v〉 = 〈v ,u〉 for all u, v ∈ Cn

(iii) 〈u, v + w〉 = 〈u, v〉+ 〈u,w〉 for all u, v ,w ∈ Cn

(iv) 〈u, αv〉 = α〈u, v〉 for all α ∈ C and all u, v ∈ Cn

For instance

〈u, v〉 = u∗

s1

s2. . .

sn

v

(for given positive real numbers s1, . . . sn) is an inner product.

Emre Mengi Week 1, Lecture 1

General Inner ProductsDefinition (Inner Product)

A function 〈·〉 : Cn × Cn → C is called an innerproduct if it satisfies the following.

(i) 〈v , v〉 > 0 for all nonzero v ∈ Cn

(ii) 〈u, v〉 = 〈v ,u〉 for all u, v ∈ Cn

(iii) 〈u, v + w〉 = 〈u, v〉+ 〈u,w〉 for all u, v ,w ∈ Cn

(iv) 〈u, αv〉 = α〈u, v〉 for all α ∈ C and all u, v ∈ Cn

For instance

〈u, v〉 = u∗

s1

s2. . .

sn

v

(for given positive real numbers s1, . . . sn) is an inner product.

Emre Mengi Week 1, Lecture 1

Orthogonality

Emre Mengi Week 1, Lecture 1

Definition

Two vectors u, v ∈ Cn are said to be orthogonal if u∗v = 0.

In this case, we use the notation u⊥v .

Examples: 121

⊥ 1−1

1

, 1

21

6⊥ 1

01

Emre Mengi Week 1, Lecture 1

Definition

Two vectors u, v ∈ Cn are said to be orthogonal if u∗v = 0.

In this case, we use the notation u⊥v .

Examples: 121

⊥ 1−1

1

, 1

21

6⊥ 1

01

Emre Mengi Week 1, Lecture 1

Definition

Two vectors u, v ∈ Cn are said to be orthogonal if u∗v = 0.

In this case, we use the notation u⊥v .

Examples: 121

⊥ 1−1

1

, 1

21

6⊥ 1

01

Emre Mengi Week 1, Lecture 1

Orthonormal Sets

The set {v (1), v (2), . . . , v (p)} is said to be orthogonal if

v (j)⊥v (k) whenever j 6= k .

The set {v (1), v (2), . . . , v (p)} is said to be orthonormal if

(i) v (j)⊥v (k) whenever j 6= k , and

(ii) ‖v (k)‖2 = 1 for k = 1, . . . ,p.

Emre Mengi Week 1, Lecture 1

Orthonormal Sets

The set {v (1), v (2), . . . , v (p)} is said to be orthogonal if

v (j)⊥v (k) whenever j 6= k .

The set {v (1), v (2), . . . , v (p)} is said to be orthonormal if

(i) v (j)⊥v (k) whenever j 6= k , and

(ii) ‖v (k)‖2 = 1 for k = 1, . . . ,p.

Emre Mengi Week 1, Lecture 1