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Mathematics – 2011/2012 - 1 -
LESSON 4
EUCLIDEAN VECTOR SPACE
Definition : Inner product or scalar product
The map
〈 〉 :ℜ n ×ℜ n → ℜ ∀ ( x , y )→〈 x , y 〉
where
X Y
y
y x y xy x y x x t n
n n n =
1 1 1 1
Properties
1. 〈 x , y 〉=〈 y , x 〉= xt^ y = ytx (symmetry)
2. 〈 x , x 〉≥ 0 ; 〈 x , x 〉= 0 ⇔ x = 0
3. 〈 k x , y 〉= k 〈 x , y 〉
4. 〈 x + x ′, y 〉=〈 x. y 〉+〈 x ′, y 〉
Definition : Euclidean vector space
It is a vector space in ℜ n in which we define the inner product.
Definition : Norm of a vector
2 2 2 2 x =+ 〈 x , x 〉=+ x 1 + x +...+ xn
From a graphic point of view the norm of a vector is the length of a vector.
Properties
1. x ≥ 0 ∀ x ∈ℜ n , x = 0 ⇔ x = 0
2. λ x = λ⋅ x , ∀λ ∈ℜ, ∀ x ∈ℜ n
3. x + y ≤ x + y (triangle inequality)
4. < x , y >≤ x ⋅ y (inequality of Schwartz)
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Mathematics – 2011/2012 - 2 -
- x n^ { }0 , x 1 x
∀ ∈ ℜ − = that is,
x
x is an unitary vector (its norm equals 1)
Remark
From (5) properties we can deduce how to normalize a vector x ≠ 0 it suffices to divide
by its norm.
Example
- x = ( 1 , 0 , 1 ) y =( 2 , 1 , 1 )
- x + y =( 3 , 1 , 2 )
- − 3 x = −( 3, 0, −3)
- < xy >= 2 + 0 + 1 = 3 ∈ℜ
- x = + < xx >=+ 1 2 + 02 + 12 =+ 2
- y = + < yy >=+ 4 + 1 + 1 =+ 6
- x + y =+ < x + y , x + y >=+ 32 + 12 + 22 =+ 14 <+ 2 + 6
- − 3 x =+ (− 3 )^2 + 02 + 32 =+ 18 = 3 2 =− 3 ⋅ x
- ) 1 2
2 (^1
, 0 ,^1
= (^1 ,^0 ,^1 ) = =+ ⋅^2 =+
x
x
Definition : Distance between two vectors
d :ℜ n ×ℜ n → ℜ^ + 2 2 ∀( x , y )→ d ( x , y )= x − y =+ ( x 1 − y 1 ) +...+( xn − yn )
Example
u = ( 1 , 2 ), v =( 3 ,− 2 ) d ( u , v )= u − v =(− 2 , 4 ) =+ (− 2 )^2 + 42 = 20
Properties
1) ∀ x , y ∈ℜ n^ , d ( x , y )≥ 0 , d ( x , y )= 0 ⇔ x = y
2) ∀ x , y ∈ℜ n^ , d ( x , y )= d ( y , x )
3) ∀ x , y , z ∈ℜ n^ , d ( x , z )≤ d ( x , y )+ d ( y , z ) (Triangle inequality)
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Mathematics – 2011/2012 - 4 -
Properties
If { x 1 ,..., xp }are orthogonal ⇒ { x (^) 1 ,..., xp }are L.I.
The set of vectors { x 1 (^) ,..., xp } ⊂ℜ n makes up an orthogonal basis for ℜ n if p = n , and they
are orthogonal.
If, moreover, x 1 = ...= xp = 1 ,that basis is orthonormal
Properties
If x is an orthogonal vector to { x (^) 1 ,..., xh }, it will be also orthogonal to any linear
combination of them.
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QUADRATIC FORMS
In this section we will see that matrices also play an important role in the study of
quadratic equations. With each quadratic equation we can associate a vector
function f ( x )= x ' Ax. Such a vector function is called a “quadratic form”.
Definition : Quadratic form
A quadratic equation in two variables x and y is an equation as:
ax^2 + 2 bxy + cy^2 = 0
Example
2
1 (^1 )
y
y f x y xy x y x x
Matrix associated with a quadratic form
11 12 1 1 12 22 2 1 1 1
1 2
t (^) n
N t N n n n MATRIXFORM (^) n OF Q (^) N N NN A A M
x x x u x u Q x f x x X A X x x x
= ∈
^
= + + → = = ⋅ ⋅ = ^ =
^
^
OFPOLYNOMIAL Q FORM
i j
n i
n nn n j ij
a x a xx a xx a x a x x a x ∑∑ a xx
= =
1 1
2 23 2 3
2 12 1 2 13 1 3 22 2
2 11 1 2 2 ...^2 ...
Change from the polynomial form to matrix form and in reverse
The elements of the principal diagonal of A are the coefficients of the quadratic terms of
the polynomial and the rest of the terms are half of values of polynomial expression.
Example
a) Given Q ( x )= Xt^ AX , Q :ℜ^3 →ℜ, such that
A the polynomial form
of Q is:
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c) 2 2
x Q x y z x y x y z y x y z Q is P S D z
= + = ^ ^ ≥ ∀ ≠ ⇒
Definition Congruent matrices
A = A^ t and B = Bt^ , A ∈ Mn , B ∈ Mn , are congruent matrices ⇔ ∃ P ∈ Mn , regular matrix,
such that:
B = Pt AP
Properties:
1. sgn A =sgn B
- Rk (^) ( A (^) ) = Rk (^) ( B )
Studying the sign of a quadratic form through its eigenvalues
First, we compute the Eigenvalues associated to matrix A. The sign of the real
quadratic form is evaluated according to the following:
Q ( x )is POSITIVE DEFINITE ⇔ λ i > 0 ∀ i = 1 , 2 ,... n
Q ( x )is POSITIVE SEMIDEFINITE ⇔ λ i > 0 and some λ i = 0
Q ( x )is NEGATIVE DEFINITE ⇔ λ i < 0 ∀ i = 1 , 2 ,... n
Q ( x )is NEGATIVE SEMIDEFINITE ⇔ λ i < 0 and some λ i = 0
Q ( x )is INDEFINITE ⇔ ∃ λ i < 0 and λ i > 0
Studying the sign of a quadratic form through its principal minors
Theorem:
Let Q ( x ) = XtAX be a quadratic form, and A = At^ ∈ Mn , let A 1 … be the principal minors of
A :
A 1 (^) = a 11 = a 11
22
12 21
11 (^2) a
a a
a A =
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Mathematics – 2011/2012 - 8 -
nn
n
n
n n
n a
a
a
a a
a a
a a A A ... ...
2
1
1 2
21 22
11 12 = =
then it can be verified:
1. Q ( x ) is P. D ⇔ Ai > 0 , ∀ i = 1 , 2 ,..., n
2. Q ( x ) is N. D .⇔ A 1 < 0 , A 2 > 0 , A 3 < 0 ,...
3. If Ai > 0, ∀ = i 1, 2,..., n and An = 0 ⇒ Q x is P S D ( )...
- If A 1 (^) < 0 , A 2 > 0 , A 3 < 0 ,K, An = 0 ⇒ Q ( x ) isN.S.D.
5. If An ≠ 0 and Qisnotdefinite ⇒ QisINDEFINITE
6. If
An = 0 and Ai ≠ 0 , ∀ i = 1 , 2 ,..., n − 1 and QisnotS. D .⇒ Qis INDEFINITE
In any other cases this method cannot study the sign.
Example
Study the sign of Q :
a) Q ( x , y , z )= 3 x^2 + 3 z^2 + 4 xy + 8 xz + 4 yz
b)
y
x Q x y xy 2
c) Q ( x , y , z )= 4 xy − 6 yz
a)
3
2
1
A A
A
A
A
Q is INDEFINITE because of (5)
b)
^ ⇒
2
1 A
A
A
The principal minors’ method doesn’t work to study the sign of Q
Q ( x , y )= − 2 y^2 ≤ 0 ∀( x , y )⇒ Q is N.S.D. ( a 11 = 0 , a 22 =− 2 < 0 )
