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Quadratic
Forms
8
...
We now ex plore the relationship between symmetric matrices and an impor
tant class of functions called quadratic forms, which are not linear
...
We shall see
in Section 8
...
Quadratic Forms
Consl
...
If
c
[
][
]
-
b/2
C
a
b/2
]
x1
...
Thus, corresponding to every symmetric matrix A, there is a quadratic form
On the other hand, given a quadratic form Q(x)
the symmetric matrix A
=
[;
b 2
�]
b 2
= axT+bx12x +ex�, we can reconstruct
by choosing (A)1
1 to be the coefficient of T
X ,
(A)12= (Ah1 to be half of the coefficient of 1
x 2x , and (A)22 to be the coefficient of x�
...
EXAMPLE 1
Determine the symmetric matrix corresponding to the quadratic form
Solution: The corresponding symmetric matrix A is
[
2
A=
-2
-
2
-1
]
Notice that we could have written axT + bx1x2 +ex� in terms of other asymmetric
matrices
...
However, we agree always to choose the symmetric matrix
for two reasons
...
Second, the choice of the symmetric matrix A allows us to apply
the special theory available for symmetric matrices
...
Definition
Quadratic Form
A quadratic form on JR
...
11
�
JR defined by
11
Q(x)
� aijXiXj
=
=
...
11, we can easily construct the co1Te
sponding symmetric matrix A by taking (A);; to be the coefficient of x and (A);j to be
�
half of the coefficient of X;Xj for i :f
...
EXAMPLE2
Q(x)
...
2 with corresponding symmetric
5/2
2
...
3 with corre-
0
-
...
2 with
0
3
0 ·
Find the quadratic form c01Tesponding to each of the following symmetric matrices
...
]
·
Find the corresponding symmetric matrix for each of the following quadratic forms
...
Q(x)
=
xi - 2x,x2 - 3�
2
...
Q(x)
=
xi + 2x� + 3x� + 4x�
+
3x1x2 - x1x3
+
4x�
�
+ x
Observe that the symmetric matrix corresponding to Q(x)
=
xi
+
2x�
+
3x�
+
4�
is in fact diagonal
...
Definition
Diagonal Form
A quadratic form Q(x) is in diagonal form if all the coefficients ajk with j * k
are equal to 0
...
EXAMPLE3
The quadratic form Q(x)
Q(x)
=
2xi - 4x1x2
+
=
3xi - 2x�
+
4 � is in diagonal form
...
Since each quadratic form has an associated symmetric matrix, we should expect
that diagonalizing the symmetric matrix should also diagonalize the quadratic form
...
EXAMPLE4
Let A = [� �]and let Q(x) =
...
Let x = Py, where
P= [ l/Yl 1/Yll Express Q(x) terms of y= [yYi2]
...
In particular, we have
+
+
...
In particular, in Example 4, we put Q(x) into diagonal form by writing it with respect to
the orthonormal basis 13={l � j�], [-1�1]}· The vector y is just the 13-coordinates
with respect to x
...
We now prove this in general
...
11• Then there is an orthonormal basis 13
of JR
...
Since A is symmetric,pwe can apply the Principal Axis Theorem to get an
orthogonal matrix P such that TAP=D = diag(;J
...
11), where ;J
...
11 are the
eigenvalues of A
...
4 that the change of coordinates matrix P from
13-coordinates to standard coordinates satisfies x=P[x]23
...
...
,
Let [X]B
�
l:J
·
Then we have
Q(x)= [Y1
...
EXAMPLES
Let Q(x)
=xi+4x1x2+x�
...
Solution:
...
matnx
...
A
The corresponding
symmetnc
1
[ ]
2
1
=
2 1
...
For
;J,1=3, we get
[
-2
2
A - '1i/ =
2 -2
An eigenvector for
For
;J,1 is v1 =
A2 = -1, we get
[il
A
An eigenvector for
;J,2 is v2 =
] [ l -1 ]
�
0
0
and a basis for the eigenspace is
- -12/=
[-i l
[; ;] [� �]
�
and a basis for the eigenspace is
Therefore, we see that A is orthogonally diagonalized by P
=
D
[� -n
Let 1
{v 1 }
...
= �
[i ]
-1
1
to
=Py
...
EXERCISE 3
Let Q(x)
= 4x1x2 -3x�
...
Classifications of Quadratic Forms
Definition
Positive Definite
Negative Definite
Indefinite
Semidefinite
A quadratic form Q(x) on JR" is
1
...
Negative definite if Q(x) < 0 for all x * 0
3
...
Positive semidefinite if Q(x) � 0 for all x
5
...
For example, we shall see in Section 8
...
Classify the quadratic forms Q1(x) = 3x
4x1x2 + x�
...
Q2(x) is indefinite since Q(l, 1) = 1>0 and Q(-1, 1 ) = - 1 < 0
...
=
Observe that classifying Q3(X) was a little more difficult than classifying Q1(X) or
Q2(X)
...
The following theorem gives us an easier way to classify a quadratic form
...
xr Ax, where A is a symmetric matrix
...
Q(x) is positive definite if and only if all eigenvalues of A are positive
...
Q(x) is negative definite if and only if all eigenvalues of A are negative
...
Q(x) is indefinite if and only if some of the eigenvalues of A are positive and
some are negative
...
By Theorem 1, there exists an orthogonal matrix P such that
where x
=
Py and /l
...
...
11 are the eigenvalues of A
...
Moreover, since
invertible
...
P
0
is orthogonal, it is
Thus we have shown that
Q(x) is positive definite if and only if all eigenvalues of A are positive
...
(a) Q(x)
=
4x� + 8x1x2 + 3x�
Solution: The symmetric matrix corresponding to Q(x) is A
2
=
[: �l
The character
istic polynomial of A is C(/l) = /l
...
Using the quadratic formula, we find that
j3
...
Hence, Q(x) is positive
the eigenvalues of A are /l = ?±
definite
...
[=� =� �]·
The
-2
1
Thus, the eigenvalues of A
-4
...
Classify the following quadratic forms
...
That is, for example, we will say a symmetric matrix A is positive definite if and only
Q(x) _xTAx is positive definite
...
if the quadratic form
EXAMPLE8
=
Classify the following symmetric matrices
...
Thus, the eigenvalues of A are 5 and 1, so A
is positive definite
...
Thus, the eigenvalues of A are
-4, 2 -
...
, so A is indefinite
...
2
Practice Problems
-2
Al Determine the quadratic form corresponding to the
given symmetric matrix
...
(ii) Express
Q(x)
in diagonal form and give the
orthogonal matrix that brings it into this form
...
xT - 3x1 x2 + x�
Q(x)= SxT - 4x1x2 + 2x�
Q(x)= -2xT + 12x1x2 + 7x�
Q(x)= xT
2x1x2 + 6x1x3 + x� + 6x2x3 -3
...
(a)A=
�[ -� �1
[ � -� -�1
[ � 1� =�1
[=; -; !]
(e)A=
-1 -2
[� � �1
(fj A=
0 0 3
-3
7
Homework Problems
Bl Determine the quadratic form corresponding to the
[! �]
given symmetric matrix
...
(d)A=
(ii) Express Q(x) in diagonal form and give the
orthogonal matrix that brings it into this form
...
7xT + 4x1x2 + 4�
Q(x)= 2x1 + 6x1x2 + 2x�
(a) Q(x)=
(b)
_
l
(b)A= -
B2 For each of the following quadratic forms Q(x),
the
[ 4 -14]
[ � -;J
B3 Classify each of the following symmetric matrices
...
S
Q(x)= 2xT-4x1x2 + 6x1x3 + 2� + 6x2x3 - 3x�
(c) Q(x)=
(e)A=
n -� j]
n -: J
[=r =� =!I
Computer Problems
Cl Classify each of the following quadratic forms with
the help of a computer
...
lxT - 0
...
2x1x4 + 2
...
6x2x3 + l
...
2x3x4 + l
...
85(xT + x� + x� + x�) - O
...
x1
...
6X1X3 + 0
...
2X2
...
6x2X4 -O
...
xr
Ax, whereA is a symmetric matrix
...
xr
Ax, whereA is a symmetric matrix
...
of the eigenvalues ofA are positive and some are
negative
...
XTAx, where A is a symmetric ma
tlix
...
...
Prove that ATA 1s
positive semidefinite
...
Prove that
(a) The diagonal entries of A are all positive
...
(c) A-1 is positive definite
...
DS A matrix B is called skew-symmetric if BT = -B
...
XS>=
n
11
I I xiY1
i=l
j=l
(b) Let G be the n x n matrix defined by gii =
(ei, e1)
...
(d) By adapting the proof of Theorem 1, show that
there is a basis 13 = {v1,
...
1
...
1,
...
...
In
and the skew-symmetric part of A to be
(1,y) = 111112 =
(a) Verify that A+ is symmetric, A- is skew
symmetric, and A= A+ +A-
...
(c) Determine expressions for typical entries (A +)ij
and (A-)iJ in terms of the entries of A
...
, wn} by
defining wi = vJ
...
...
...
C-coordmates, so that x = x1 W1 +
+ xnwn
...
)
D6 In this problem, we show that general inner prod
ucts on JR11 are not different in interesting ways from
the standard inner product
...
, e,1} be the standard basis
...
(1,y) = x�y� +
·
·
·
+
x�y�
Thus, with respect to the inner product(,), C is
an orthonormal basis, and in C-coordinates, the
inner product of two vectors looks just like the
standard dot product