Quadratic Forms (Reduction and Classification of Quadratic Forms) - CSIR-NET Mathematical Sciences | Mathematics for IIT JAM, GATE, CSIR NET, UGC NET PDF Download

 Page 1


1 Quadratic Forms
A quadratic function f : R! R has the form f(x) = a¢x
2
. Generalization
of this notion to two variables is the quadratic form
Q(x
1
;x
2
)=a
11
x
2
1
+a
12
x
1
x
2
+a
21
x
2
x
1
+a
22
x
2
2
:
Here each term has degree 2 (the sum of exponents is 2 for all summands).
A quadratic form of three variables looks as
f(x
1
;x
2
;x
3
)=
a
11
x
2
1
+a
12
x
1
x
2
+a
13
x
1
x
3
+
a
21
x
2
x
1
+a
22
x
2
2
+a
23
x
2
x
3
+
a
31
x
1
x
3
+a
32
x
3
x
2
+a
33
x
2
3
:
Ageneralquadraticformofnvariablesisareal-valuedfunctionQ:R
n
!
R of the form
Q(x
1
;x
2
;:::;x
n
)= a
11
x
2
1
+ a
12
x
1
x
2
+ ::: + a
1n
x
1
x
n
+
a
21
x
2
x
1
+ a
22
x
2
2
+ ::: + a
2n
x
2
x
n
+
::: ::: ::: :::
a
n1
x
n
x
1
+ a
n2
x
n
x
2
+ ::: + a
nn
x
2
n
In short Q(x
1
;x
2
;:::;x
n
)=
P
n
i;j
a
ij
x
i
x
j
.
As we see a quadratic form is determined by the matrix
A=
0
B
@
a
11
::: a
1n
::::::::::::::
a
n1
::: a
nn
1
C
A:
1.1 Matrix Representation of Quadratic Forms
Let Q(x
1
;x
2
;:::;x
n
)=
P
n
i;j
a
ij
x
i
x
j
be a quadratic form with matrix A. Easy
to see that
Q(x
1
;:::;x
n
)=(x
1
;:::;x
n
)¢
0
B
@
a
11
::: a
1n
::::::::::::::
a
n1
::: a
nn
1
C
A¢
0
B
@
x
1
::
x
n
1
C
A:
Page 2


1 Quadratic Forms
A quadratic function f : R! R has the form f(x) = a¢x
2
. Generalization
of this notion to two variables is the quadratic form
Q(x
1
;x
2
)=a
11
x
2
1
+a
12
x
1
x
2
+a
21
x
2
x
1
+a
22
x
2
2
:
Here each term has degree 2 (the sum of exponents is 2 for all summands).
A quadratic form of three variables looks as
f(x
1
;x
2
;x
3
)=
a
11
x
2
1
+a
12
x
1
x
2
+a
13
x
1
x
3
+
a
21
x
2
x
1
+a
22
x
2
2
+a
23
x
2
x
3
+
a
31
x
1
x
3
+a
32
x
3
x
2
+a
33
x
2
3
:
Ageneralquadraticformofnvariablesisareal-valuedfunctionQ:R
n
!
R of the form
Q(x
1
;x
2
;:::;x
n
)= a
11
x
2
1
+ a
12
x
1
x
2
+ ::: + a
1n
x
1
x
n
+
a
21
x
2
x
1
+ a
22
x
2
2
+ ::: + a
2n
x
2
x
n
+
::: ::: ::: :::
a
n1
x
n
x
1
+ a
n2
x
n
x
2
+ ::: + a
nn
x
2
n
In short Q(x
1
;x
2
;:::;x
n
)=
P
n
i;j
a
ij
x
i
x
j
.
As we see a quadratic form is determined by the matrix
A=
0
B
@
a
11
::: a
1n
::::::::::::::
a
n1
::: a
nn
1
C
A:
1.1 Matrix Representation of Quadratic Forms
Let Q(x
1
;x
2
;:::;x
n
)=
P
n
i;j
a
ij
x
i
x
j
be a quadratic form with matrix A. Easy
to see that
Q(x
1
;:::;x
n
)=(x
1
;:::;x
n
)¢
0
B
@
a
11
::: a
1n
::::::::::::::
a
n1
::: a
nn
1
C
A¢
0
B
@
x
1
::
x
n
1
C
A:
Equivalently Q(x)=x
T
¢A¢x.
Example. The quadratic form Q(x
1
;x
2
;x
3
) = 5x
2
1
¡ 10x
1
x
2
+ x
2
2
whose
symmetric matrix is A=
Ã
5 ¡5
¡5 1
!
is the product of three matrices
(x
1
;x
2
;x
3
)¢
Ã
5 ¡5
¡5 1
!
¢
0
B
@
x
1
x
2
x
3
1
C
A:
1.1.1 Symmetrization of matrix
The quadratic form Q(x
1
;x
2
;x
3
)=5x
2
1
¡10x
1
x
2
+x
2
2
can be represented, for
example, by the following 2£2 matrices
Ã
5 ¡2
¡8 1
!
;
Ã
5 ¡3
¡7 1
!
;
Ã
5 ¡5
¡5 1
!
the last one is symmetric: a
ij
=a
ji
.
Theorem 1 Any quadratic form can be represented by symmetric matrix.
Indeed, if a
ij
6= a
ji
we replace them by new a
0
ij
= a
0
ji
=
a
ij
+a
ji
2
, this does
not change the corresponding quadratic form.
Generally, one can ?nd symmetrization A
0
of a matrix A by A
0
=
A+A
T
2
.
1.2 De?niteness of Quadratic Forms
A quadratic form of one variable is just a quadratic function Q(x)=a¢x
2
.
If a> 0 then Q(x)>0 for each nonzero x.
If a< 0 then Q(x)<0 for each nonzero x.
Sothesignofthecoe?cientadeterminesthesignofonevariable quadratic
form.
The notion of de?niteness described bellow generalizes this phenomenon
for multivariable quadratic forms.
1.2.1 Generic Examples
The quadratic form Q(x;y) = x
2
+ y
2
is positive for all nonzero (that is
(x;y)6=(0;0)) arguments (x;y). Such forms are called positive de?nite.
The quadratic form Q(x;y) =¡x
2
¡y
2
is negative for all nonzero argu-
ments (x;y). Such forms are called negative de?nite.
The quadratic form Q(x;y) = (x¡y)
2
is nonnegative. This means that
Q(x;y) = (x¡y)
2
is either positive or zero for nonzero arguments. Such
forms are called positive semide?nite.
Page 3


1 Quadratic Forms
A quadratic function f : R! R has the form f(x) = a¢x
2
. Generalization
of this notion to two variables is the quadratic form
Q(x
1
;x
2
)=a
11
x
2
1
+a
12
x
1
x
2
+a
21
x
2
x
1
+a
22
x
2
2
:
Here each term has degree 2 (the sum of exponents is 2 for all summands).
A quadratic form of three variables looks as
f(x
1
;x
2
;x
3
)=
a
11
x
2
1
+a
12
x
1
x
2
+a
13
x
1
x
3
+
a
21
x
2
x
1
+a
22
x
2
2
+a
23
x
2
x
3
+
a
31
x
1
x
3
+a
32
x
3
x
2
+a
33
x
2
3
:
Ageneralquadraticformofnvariablesisareal-valuedfunctionQ:R
n
!
R of the form
Q(x
1
;x
2
;:::;x
n
)= a
11
x
2
1
+ a
12
x
1
x
2
+ ::: + a
1n
x
1
x
n
+
a
21
x
2
x
1
+ a
22
x
2
2
+ ::: + a
2n
x
2
x
n
+
::: ::: ::: :::
a
n1
x
n
x
1
+ a
n2
x
n
x
2
+ ::: + a
nn
x
2
n
In short Q(x
1
;x
2
;:::;x
n
)=
P
n
i;j
a
ij
x
i
x
j
.
As we see a quadratic form is determined by the matrix
A=
0
B
@
a
11
::: a
1n
::::::::::::::
a
n1
::: a
nn
1
C
A:
1.1 Matrix Representation of Quadratic Forms
Let Q(x
1
;x
2
;:::;x
n
)=
P
n
i;j
a
ij
x
i
x
j
be a quadratic form with matrix A. Easy
to see that
Q(x
1
;:::;x
n
)=(x
1
;:::;x
n
)¢
0
B
@
a
11
::: a
1n
::::::::::::::
a
n1
::: a
nn
1
C
A¢
0
B
@
x
1
::
x
n
1
C
A:
Equivalently Q(x)=x
T
¢A¢x.
Example. The quadratic form Q(x
1
;x
2
;x
3
) = 5x
2
1
¡ 10x
1
x
2
+ x
2
2
whose
symmetric matrix is A=
Ã
5 ¡5
¡5 1
!
is the product of three matrices
(x
1
;x
2
;x
3
)¢
Ã
5 ¡5
¡5 1
!
¢
0
B
@
x
1
x
2
x
3
1
C
A:
1.1.1 Symmetrization of matrix
The quadratic form Q(x
1
;x
2
;x
3
)=5x
2
1
¡10x
1
x
2
+x
2
2
can be represented, for
example, by the following 2£2 matrices
Ã
5 ¡2
¡8 1
!
;
Ã
5 ¡3
¡7 1
!
;
Ã
5 ¡5
¡5 1
!
the last one is symmetric: a
ij
=a
ji
.
Theorem 1 Any quadratic form can be represented by symmetric matrix.
Indeed, if a
ij
6= a
ji
we replace them by new a
0
ij
= a
0
ji
=
a
ij
+a
ji
2
, this does
not change the corresponding quadratic form.
Generally, one can ?nd symmetrization A
0
of a matrix A by A
0
=
A+A
T
2
.
1.2 De?niteness of Quadratic Forms
A quadratic form of one variable is just a quadratic function Q(x)=a¢x
2
.
If a> 0 then Q(x)>0 for each nonzero x.
If a< 0 then Q(x)<0 for each nonzero x.
Sothesignofthecoe?cientadeterminesthesignofonevariable quadratic
form.
The notion of de?niteness described bellow generalizes this phenomenon
for multivariable quadratic forms.
1.2.1 Generic Examples
The quadratic form Q(x;y) = x
2
+ y
2
is positive for all nonzero (that is
(x;y)6=(0;0)) arguments (x;y). Such forms are called positive de?nite.
The quadratic form Q(x;y) =¡x
2
¡y
2
is negative for all nonzero argu-
ments (x;y). Such forms are called negative de?nite.
The quadratic form Q(x;y) = (x¡y)
2
is nonnegative. This means that
Q(x;y) = (x¡y)
2
is either positive or zero for nonzero arguments. Such
forms are called positive semide?nite.
The quadratic form Q(x;y)=¡(x¡y)
2
is nonpositive. This means that
Q(x;y) = (x¡y)
2
is either negative or zero for nonzero arguments. Such
forms are called negative semide?nite.
The quadratic form Q(x;y) = x
2
¡ y
2
is called inde?nite since it can
take both positive and negative values, for example Q(3;1) = 9¡1 = 8 >
0; Q(1;3)=1¡9=¡8<0.
1.2.2 De?niteness
De?nition. A quadratic form Q(x) = x
T
¢A¢x (equivalently a symmetric
matrix A) is
(a) positive de?nite if Q(x)>0 for all x6=02R
n
;
(b) positive semide?nite if Q(x)¸0 for all x6=02R
n
;
(c) negative de?nite if Q(x)<0 for all x6=02R
n
;
(d) negative semide?nite if Q(x)·0 for all x6=02R
n
;
(e) inde?nite if Q(x)>0 for some x and Q(x)<0 for some other x.
1.2.3 De?niteness and Optimality
Determiningthede?nitenessofquadraticformQisequivalenttodetermining
wether x=0 is max, min or neither. Particularly:
If Q is positive de?nite then x=0 is global maximum;
If Q is negative de?nite then x=0 is global minimum.
1.2.4 De?niteness of 2 Variable Quadratic Form
Let Q(x
1
;x
2
) = ax
2
1
+ 2bx
1
x
2
+ cx
2
2
= (x
1
;x
2
)¢
Ã
a b
b c
!
¢
Ã
x
1
x
2
!
be a 2
variable quadratic form.
Here A =
Ã
a b
b c
!
is the symmetric matrix of the quadratic form. The
determinant
¯
¯
¯
¯
¯
a b
b c
¯
¯
¯
¯
¯
=ac¡b
2
is called discriminant of Q.
Easy to see that
ax
2
1
+2bx
1
x
2
+cx
2
2
=a(x
1
+
b
a
x
2
)
2
+
ac¡b
2
a
x
2
2
:
Let us use the notation D
1
= a; D
2
= ac¡ b
2
. Actually D
1
and D
2
are
leading principal minors of A. Note that there exists one more principal
(non leading) minor (of degree 1) D
0
1
=c.
Then
Q(x
1
;x
2
)=D
1
(x
1
+
b
a
x
2
)
2
+
D
2
D
1
x
2
2
:
From this expression we obtain:
Page 4


1 Quadratic Forms
A quadratic function f : R! R has the form f(x) = a¢x
2
. Generalization
of this notion to two variables is the quadratic form
Q(x
1
;x
2
)=a
11
x
2
1
+a
12
x
1
x
2
+a
21
x
2
x
1
+a
22
x
2
2
:
Here each term has degree 2 (the sum of exponents is 2 for all summands).
A quadratic form of three variables looks as
f(x
1
;x
2
;x
3
)=
a
11
x
2
1
+a
12
x
1
x
2
+a
13
x
1
x
3
+
a
21
x
2
x
1
+a
22
x
2
2
+a
23
x
2
x
3
+
a
31
x
1
x
3
+a
32
x
3
x
2
+a
33
x
2
3
:
Ageneralquadraticformofnvariablesisareal-valuedfunctionQ:R
n
!
R of the form
Q(x
1
;x
2
;:::;x
n
)= a
11
x
2
1
+ a
12
x
1
x
2
+ ::: + a
1n
x
1
x
n
+
a
21
x
2
x
1
+ a
22
x
2
2
+ ::: + a
2n
x
2
x
n
+
::: ::: ::: :::
a
n1
x
n
x
1
+ a
n2
x
n
x
2
+ ::: + a
nn
x
2
n
In short Q(x
1
;x
2
;:::;x
n
)=
P
n
i;j
a
ij
x
i
x
j
.
As we see a quadratic form is determined by the matrix
A=
0
B
@
a
11
::: a
1n
::::::::::::::
a
n1
::: a
nn
1
C
A:
1.1 Matrix Representation of Quadratic Forms
Let Q(x
1
;x
2
;:::;x
n
)=
P
n
i;j
a
ij
x
i
x
j
be a quadratic form with matrix A. Easy
to see that
Q(x
1
;:::;x
n
)=(x
1
;:::;x
n
)¢
0
B
@
a
11
::: a
1n
::::::::::::::
a
n1
::: a
nn
1
C
A¢
0
B
@
x
1
::
x
n
1
C
A:
Equivalently Q(x)=x
T
¢A¢x.
Example. The quadratic form Q(x
1
;x
2
;x
3
) = 5x
2
1
¡ 10x
1
x
2
+ x
2
2
whose
symmetric matrix is A=
Ã
5 ¡5
¡5 1
!
is the product of three matrices
(x
1
;x
2
;x
3
)¢
Ã
5 ¡5
¡5 1
!
¢
0
B
@
x
1
x
2
x
3
1
C
A:
1.1.1 Symmetrization of matrix
The quadratic form Q(x
1
;x
2
;x
3
)=5x
2
1
¡10x
1
x
2
+x
2
2
can be represented, for
example, by the following 2£2 matrices
Ã
5 ¡2
¡8 1
!
;
Ã
5 ¡3
¡7 1
!
;
Ã
5 ¡5
¡5 1
!
the last one is symmetric: a
ij
=a
ji
.
Theorem 1 Any quadratic form can be represented by symmetric matrix.
Indeed, if a
ij
6= a
ji
we replace them by new a
0
ij
= a
0
ji
=
a
ij
+a
ji
2
, this does
not change the corresponding quadratic form.
Generally, one can ?nd symmetrization A
0
of a matrix A by A
0
=
A+A
T
2
.
1.2 De?niteness of Quadratic Forms
A quadratic form of one variable is just a quadratic function Q(x)=a¢x
2
.
If a> 0 then Q(x)>0 for each nonzero x.
If a< 0 then Q(x)<0 for each nonzero x.
Sothesignofthecoe?cientadeterminesthesignofonevariable quadratic
form.
The notion of de?niteness described bellow generalizes this phenomenon
for multivariable quadratic forms.
1.2.1 Generic Examples
The quadratic form Q(x;y) = x
2
+ y
2
is positive for all nonzero (that is
(x;y)6=(0;0)) arguments (x;y). Such forms are called positive de?nite.
The quadratic form Q(x;y) =¡x
2
¡y
2
is negative for all nonzero argu-
ments (x;y). Such forms are called negative de?nite.
The quadratic form Q(x;y) = (x¡y)
2
is nonnegative. This means that
Q(x;y) = (x¡y)
2
is either positive or zero for nonzero arguments. Such
forms are called positive semide?nite.
The quadratic form Q(x;y)=¡(x¡y)
2
is nonpositive. This means that
Q(x;y) = (x¡y)
2
is either negative or zero for nonzero arguments. Such
forms are called negative semide?nite.
The quadratic form Q(x;y) = x
2
¡ y
2
is called inde?nite since it can
take both positive and negative values, for example Q(3;1) = 9¡1 = 8 >
0; Q(1;3)=1¡9=¡8<0.
1.2.2 De?niteness
De?nition. A quadratic form Q(x) = x
T
¢A¢x (equivalently a symmetric
matrix A) is
(a) positive de?nite if Q(x)>0 for all x6=02R
n
;
(b) positive semide?nite if Q(x)¸0 for all x6=02R
n
;
(c) negative de?nite if Q(x)<0 for all x6=02R
n
;
(d) negative semide?nite if Q(x)·0 for all x6=02R
n
;
(e) inde?nite if Q(x)>0 for some x and Q(x)<0 for some other x.
1.2.3 De?niteness and Optimality
Determiningthede?nitenessofquadraticformQisequivalenttodetermining
wether x=0 is max, min or neither. Particularly:
If Q is positive de?nite then x=0 is global maximum;
If Q is negative de?nite then x=0 is global minimum.
1.2.4 De?niteness of 2 Variable Quadratic Form
Let Q(x
1
;x
2
) = ax
2
1
+ 2bx
1
x
2
+ cx
2
2
= (x
1
;x
2
)¢
Ã
a b
b c
!
¢
Ã
x
1
x
2
!
be a 2
variable quadratic form.
Here A =
Ã
a b
b c
!
is the symmetric matrix of the quadratic form. The
determinant
¯
¯
¯
¯
¯
a b
b c
¯
¯
¯
¯
¯
=ac¡b
2
is called discriminant of Q.
Easy to see that
ax
2
1
+2bx
1
x
2
+cx
2
2
=a(x
1
+
b
a
x
2
)
2
+
ac¡b
2
a
x
2
2
:
Let us use the notation D
1
= a; D
2
= ac¡ b
2
. Actually D
1
and D
2
are
leading principal minors of A. Note that there exists one more principal
(non leading) minor (of degree 1) D
0
1
=c.
Then
Q(x
1
;x
2
)=D
1
(x
1
+
b
a
x
2
)
2
+
D
2
D
1
x
2
2
:
From this expression we obtain:
1. If D
1
> 0 and D
2
> 0 then the form is of x
2
+y
2
type, so it is positive
de?nite;
2. If D
1
< 0 and D
2
> 0 then the form is of¡x
2
¡y
2
type, so it is negative
de?nite;
3. If D
1
>0 and D
2
<0 then the form is of x
2
¡y
2
type, so it is inde?nite;
If D
1
< 0 and D
2
< 0 then the form is of ¡x
2
+y
2
type, so it is also
inde?nite;
Thus if D
2
<0 then the form is inde?nite.
Semide?niteness depends not only on leading principal minors D
1
; D
2
but also on all principal minors, in this case on D
0
1
=c too.
4. If D
1
¸0; D
0
1
¸0 and D
2
¸0 then the form is positive semide?nite.
NotethatonlyD
1
¸0andD
2
¸0isnotenough,theadditionalcondition
D
0
1
¸0 here is absolutely necessary: consider the form Q(x
1
;x
2
)=¡x
2
2
with
a = 0; b = 0; c =¡1, here D
1
= a¸ 0; D
2
= ac¡b
2
¸ 0, nevertheless the
form is not positive semidie?nite.
5. If D
1
·0; D
0
1
·0 and D
2
¸0 then the form is negative semide?nite.
NotethatonlyD
1
·0andD
2
¸0isnotenough,theadditionalcondition
D
0
1
·0 again is absolutely necessary: consider the form Q(x
1
;x
2
)=x
2
2
with
a = 0; b = 0; c = 1, here D
1
= a· 0; D
2
= ac¡b
2
¸ 0, nevertheless the
form is not negative semidie?nite.
1.2.5 De?niteness of 3 Variable Quadratic Form
Let us start with the following
Example. Q(x
1
;x
2
;x
3
) = x
2
1
+2x
2
2
¡7x
2
3
¡4x
1
x
2
+8x
1
x
3
. The symmetric
matrix of this quadratic form is
0
B
@
1 ¡2 4
¡2 2 0
4 0 ¡7
1
C
A:
The leading principal minors of this matrix are
jD
1
j=
¯
¯
¯ 1
¯
¯
¯=1; jD
2
j=
¯
¯
¯
¯
¯
1 ¡2
¡2 2
¯
¯
¯
¯
¯
=¡2; jD
3
j=
¯
¯
¯
¯
¯
¯
¯
1 ¡2 4
¡2 2 0
4 0 ¡7
¯
¯
¯
¯
¯
¯
¯
=¡18:
Page 5


1 Quadratic Forms
A quadratic function f : R! R has the form f(x) = a¢x
2
. Generalization
of this notion to two variables is the quadratic form
Q(x
1
;x
2
)=a
11
x
2
1
+a
12
x
1
x
2
+a
21
x
2
x
1
+a
22
x
2
2
:
Here each term has degree 2 (the sum of exponents is 2 for all summands).
A quadratic form of three variables looks as
f(x
1
;x
2
;x
3
)=
a
11
x
2
1
+a
12
x
1
x
2
+a
13
x
1
x
3
+
a
21
x
2
x
1
+a
22
x
2
2
+a
23
x
2
x
3
+
a
31
x
1
x
3
+a
32
x
3
x
2
+a
33
x
2
3
:
Ageneralquadraticformofnvariablesisareal-valuedfunctionQ:R
n
!
R of the form
Q(x
1
;x
2
;:::;x
n
)= a
11
x
2
1
+ a
12
x
1
x
2
+ ::: + a
1n
x
1
x
n
+
a
21
x
2
x
1
+ a
22
x
2
2
+ ::: + a
2n
x
2
x
n
+
::: ::: ::: :::
a
n1
x
n
x
1
+ a
n2
x
n
x
2
+ ::: + a
nn
x
2
n
In short Q(x
1
;x
2
;:::;x
n
)=
P
n
i;j
a
ij
x
i
x
j
.
As we see a quadratic form is determined by the matrix
A=
0
B
@
a
11
::: a
1n
::::::::::::::
a
n1
::: a
nn
1
C
A:
1.1 Matrix Representation of Quadratic Forms
Let Q(x
1
;x
2
;:::;x
n
)=
P
n
i;j
a
ij
x
i
x
j
be a quadratic form with matrix A. Easy
to see that
Q(x
1
;:::;x
n
)=(x
1
;:::;x
n
)¢
0
B
@
a
11
::: a
1n
::::::::::::::
a
n1
::: a
nn
1
C
A¢
0
B
@
x
1
::
x
n
1
C
A:
Equivalently Q(x)=x
T
¢A¢x.
Example. The quadratic form Q(x
1
;x
2
;x
3
) = 5x
2
1
¡ 10x
1
x
2
+ x
2
2
whose
symmetric matrix is A=
Ã
5 ¡5
¡5 1
!
is the product of three matrices
(x
1
;x
2
;x
3
)¢
Ã
5 ¡5
¡5 1
!
¢
0
B
@
x
1
x
2
x
3
1
C
A:
1.1.1 Symmetrization of matrix
The quadratic form Q(x
1
;x
2
;x
3
)=5x
2
1
¡10x
1
x
2
+x
2
2
can be represented, for
example, by the following 2£2 matrices
Ã
5 ¡2
¡8 1
!
;
Ã
5 ¡3
¡7 1
!
;
Ã
5 ¡5
¡5 1
!
the last one is symmetric: a
ij
=a
ji
.
Theorem 1 Any quadratic form can be represented by symmetric matrix.
Indeed, if a
ij
6= a
ji
we replace them by new a
0
ij
= a
0
ji
=
a
ij
+a
ji
2
, this does
not change the corresponding quadratic form.
Generally, one can ?nd symmetrization A
0
of a matrix A by A
0
=
A+A
T
2
.
1.2 De?niteness of Quadratic Forms
A quadratic form of one variable is just a quadratic function Q(x)=a¢x
2
.
If a> 0 then Q(x)>0 for each nonzero x.
If a< 0 then Q(x)<0 for each nonzero x.
Sothesignofthecoe?cientadeterminesthesignofonevariable quadratic
form.
The notion of de?niteness described bellow generalizes this phenomenon
for multivariable quadratic forms.
1.2.1 Generic Examples
The quadratic form Q(x;y) = x
2
+ y
2
is positive for all nonzero (that is
(x;y)6=(0;0)) arguments (x;y). Such forms are called positive de?nite.
The quadratic form Q(x;y) =¡x
2
¡y
2
is negative for all nonzero argu-
ments (x;y). Such forms are called negative de?nite.
The quadratic form Q(x;y) = (x¡y)
2
is nonnegative. This means that
Q(x;y) = (x¡y)
2
is either positive or zero for nonzero arguments. Such
forms are called positive semide?nite.
The quadratic form Q(x;y)=¡(x¡y)
2
is nonpositive. This means that
Q(x;y) = (x¡y)
2
is either negative or zero for nonzero arguments. Such
forms are called negative semide?nite.
The quadratic form Q(x;y) = x
2
¡ y
2
is called inde?nite since it can
take both positive and negative values, for example Q(3;1) = 9¡1 = 8 >
0; Q(1;3)=1¡9=¡8<0.
1.2.2 De?niteness
De?nition. A quadratic form Q(x) = x
T
¢A¢x (equivalently a symmetric
matrix A) is
(a) positive de?nite if Q(x)>0 for all x6=02R
n
;
(b) positive semide?nite if Q(x)¸0 for all x6=02R
n
;
(c) negative de?nite if Q(x)<0 for all x6=02R
n
;
(d) negative semide?nite if Q(x)·0 for all x6=02R
n
;
(e) inde?nite if Q(x)>0 for some x and Q(x)<0 for some other x.
1.2.3 De?niteness and Optimality
Determiningthede?nitenessofquadraticformQisequivalenttodetermining
wether x=0 is max, min or neither. Particularly:
If Q is positive de?nite then x=0 is global maximum;
If Q is negative de?nite then x=0 is global minimum.
1.2.4 De?niteness of 2 Variable Quadratic Form
Let Q(x
1
;x
2
) = ax
2
1
+ 2bx
1
x
2
+ cx
2
2
= (x
1
;x
2
)¢
Ã
a b
b c
!
¢
Ã
x
1
x
2
!
be a 2
variable quadratic form.
Here A =
Ã
a b
b c
!
is the symmetric matrix of the quadratic form. The
determinant
¯
¯
¯
¯
¯
a b
b c
¯
¯
¯
¯
¯
=ac¡b
2
is called discriminant of Q.
Easy to see that
ax
2
1
+2bx
1
x
2
+cx
2
2
=a(x
1
+
b
a
x
2
)
2
+
ac¡b
2
a
x
2
2
:
Let us use the notation D
1
= a; D
2
= ac¡ b
2
. Actually D
1
and D
2
are
leading principal minors of A. Note that there exists one more principal
(non leading) minor (of degree 1) D
0
1
=c.
Then
Q(x
1
;x
2
)=D
1
(x
1
+
b
a
x
2
)
2
+
D
2
D
1
x
2
2
:
From this expression we obtain:
1. If D
1
> 0 and D
2
> 0 then the form is of x
2
+y
2
type, so it is positive
de?nite;
2. If D
1
< 0 and D
2
> 0 then the form is of¡x
2
¡y
2
type, so it is negative
de?nite;
3. If D
1
>0 and D
2
<0 then the form is of x
2
¡y
2
type, so it is inde?nite;
If D
1
< 0 and D
2
< 0 then the form is of ¡x
2
+y
2
type, so it is also
inde?nite;
Thus if D
2
<0 then the form is inde?nite.
Semide?niteness depends not only on leading principal minors D
1
; D
2
but also on all principal minors, in this case on D
0
1
=c too.
4. If D
1
¸0; D
0
1
¸0 and D
2
¸0 then the form is positive semide?nite.
NotethatonlyD
1
¸0andD
2
¸0isnotenough,theadditionalcondition
D
0
1
¸0 here is absolutely necessary: consider the form Q(x
1
;x
2
)=¡x
2
2
with
a = 0; b = 0; c =¡1, here D
1
= a¸ 0; D
2
= ac¡b
2
¸ 0, nevertheless the
form is not positive semidie?nite.
5. If D
1
·0; D
0
1
·0 and D
2
¸0 then the form is negative semide?nite.
NotethatonlyD
1
·0andD
2
¸0isnotenough,theadditionalcondition
D
0
1
·0 again is absolutely necessary: consider the form Q(x
1
;x
2
)=x
2
2
with
a = 0; b = 0; c = 1, here D
1
= a· 0; D
2
= ac¡b
2
¸ 0, nevertheless the
form is not negative semidie?nite.
1.2.5 De?niteness of 3 Variable Quadratic Form
Let us start with the following
Example. Q(x
1
;x
2
;x
3
) = x
2
1
+2x
2
2
¡7x
2
3
¡4x
1
x
2
+8x
1
x
3
. The symmetric
matrix of this quadratic form is
0
B
@
1 ¡2 4
¡2 2 0
4 0 ¡7
1
C
A:
The leading principal minors of this matrix are
jD
1
j=
¯
¯
¯ 1
¯
¯
¯=1; jD
2
j=
¯
¯
¯
¯
¯
1 ¡2
¡2 2
¯
¯
¯
¯
¯
=¡2; jD
3
j=
¯
¯
¯
¯
¯
¯
¯
1 ¡2 4
¡2 2 0
4 0 ¡7
¯
¯
¯
¯
¯
¯
¯
=¡18:
Now look:
Q(x
1
;x
2
;x
3
)=x
2
1
+2x
2
2
¡7x
2
3
¡4x
1
x
2
+8x
1
x
3
=
x
2
1
¡4x
1
x
2
+8x
1
x
3
+2x
2
2
¡7x
2
3
=x
2
1
¡4x
1
(x
2
¡2x
3
)+2x
2
2
¡7x
2
3
=
[x
2
1
¡4x
1
(x
2
¡2x
3
)+4(x
2
¡2x
3
)¡4(x
2
¡2x
3
)]+2x
2
2
¡7x
2
3
=
[x
1
¡2x
2
+4x
3
]
2
¡2x
2
2
¡16x
2
x
3
¡23x
2
3
=
[x
1
¡2x
2
+4x
3
]
2
¡2(x
2
2
¡8x
2
x
3
)¡23x
2
3
=
[x
1
¡2x
2
+4x
3
]
2
¡2[x
2
2
¡8x
2
x
3
+16x
2
3
¡16x
2
3
]¡23x
2
3
=
[x
1
¡2x
2
+4x
3
]
2
¡2[x
2
¡4x
3
]
2
¡16x
2
3
)¡23x
2
3
=
[x
1
¡2x
2
+4x
3
]
2
¡2[x
2
¡4x
3
]
2
+32x
2
3
¡23x
2
3
=
[x
1
¡2x
2
+4x
3
]
2
¡2[x
2
¡4x
3
]
2
+9x
2
3
=
jD
1
jl
2
1
+
D
2
D
1
l
2
+
D
3
D
2
l
2
3
;
where
l
1
=x
1
¡2x
2
+4x
3
;
l
2
= x
2
¡4x
3
;
l
3
= x
3
:
That is (l
1
;l
2
;l
3
) are linear combinations of (x
1
;x
2
;x
3
). More precisely
0
B
@
l
1
l
2
l
3
1
C
A=
0
B
@
1 ¡2 4
0 1 ¡4
0 0 1
1
C
A¢
0
B
@
x
1
x
2
x
3
1
C
A
where
P =
0
B
@
1 ¡2 4
0 1 ¡4
0 0 1
1
C
A
is a nonsingular matrix (changing variables).
Now turn to general 3 variable quadratic form
Q(x
1
;x
2
;x
3
)=(x
1
;x
2
;x
3
)¢
0
B
@
a
11
a
12
a
13
a
21
a
22
a
23
a
31
a
32
a
33
1
C
A¢
0
B
@
x
1
x
2
x
3
1
C
A:
The following three determinants
jD
1
j=
¯
¯
¯ a
11
¯
¯
¯; jD
2
j=
¯
¯
¯
¯
¯
a
11
a
12
a
21
a
22
¯
¯
¯
¯
¯
; jD
3
j=
¯
¯
¯
¯
¯
¯
¯
a
11
a
12
a
13
a
21
a
22
a
23
a
31
a
32
a
33
¯
¯
¯
¯
¯
¯
¯
are leading principal minors.
It is possible to show that, as in 2 variable case, if jD
1
j 6= 0; jD
2
j 6= 0,
then
Q(x
1
;x
2
;x
3
)=jD
1
jl
2
1
+
jD
2
j
jD
1
j
l
2
2
+
jD
3
j
jD
2
j
l
2
3