March 14, 2022

I.4 QUADRATIC FORMS

1. Definition and matrix associated to a quadratic form

Definition 1.1. A polynomial of n variables p(x1 , . . . , xn ) is homogeneous of degree m if

and only if
p(λx1 , . . . , λxn ) = λm p(x1 , . . . , xn ),
for all λ ∈ R.

Definition 1.2. A quadratic form Q : Rn → R is an homogeneous polynomial of degree 2,

that is
n
X
Q(x1 , . . . , xn ) = bij xi xj ,
i,j=1
b +bji
where bij ∈ R, for all i, j ∈ {1, . . . , n}. Note that, defining aij = ij 2 ,
X X
Q(x1 , . . . , xn ) = aij x2i + 2 aij xi xj .
i=1 i,j=1
j>i

Definition 1.3. The matrix associated to the quadratic form Q : Rn → R is

 
a11 · · · a1n
 . .. .. 
 ..
 . .
. 
a1n · · · ann

Note that   
a11 · · · a1n x1
 . .. ..   . 
Q(x1 , . . . , xn ) = (x1 , . . . , xn )  . .  .
 . . 
 .


a1n · · · ann xn
and that the matrix A is symmetric.

Proposition 1.4. Given a quadratic form Q : Rn → R, there is a unique symmetric matrix

such that
Q(x) = xt Ax,
for all x ∈ Rn . Reciprocally, given a symmetric matrix A ∈ Mn×n , there is a unique
quadratic form Q : Rn → R such that

Q(x) = xt Ax,

for all x ∈ Rn .
1
2 I.4 QUADRATIC FORMS

1.1. Properties. Let Q : Rn → R be a quadratic form, let x ∈ Rn and λ ∈ R.

(1) Q(0) = 0.

(2) Q(λx) = λ2 Q(x).

(3) Q(−x) = Q(x).

2. Classification of quadratic forms

Definition 2.1. Let Q : Rn → R be a quadratic form.

(1) Q is positive definite if and only if

Q(x) > 0,

for all x 6= 0.

(2) Q negative definite if and only if

Q(x) < 0,

for all x 6= 0.

(3) Q is positive semidefinite if and only if

Q(x) ≥ 0,

for all x ∈ Rn and there is y ∈ Rn non-null such that

Q(y) = 0.

(4) Q is negative semidefinite if and only if

Q(x) ≤ 0,

for all x ∈ Rn and there is z ∈ Rn non-null such that

Q(z) = 0.

(5) Q is indefinite if and only if there are y, z ∈ Rn , such that

Q(y) > 0, Q(z) < 0.

Note that any non null quadratic form is of one and only one of the classes above. The null
quadratic form is a special case.
 I.4 QUADRATIC FORMS 3

Theorem 2.2. For every quadratic form Q : Rm → R, there is a change of variable y = P x,

with P orthonormal, such that

Q(y) = λ1 y12 + · · · + λn yn2 .

Here, λ1 , . . . , λn are eigenvalues of the matrix A associated to Q (possibly repeated), P is a

matrix such that A = P t DP , and D is a diagonal matrix associated to A.

It turns out that it suffices to know the sign of the eigenvalues of A for classifying Q.

Theorem 2.3 (Criterion of the proper values). Let Q : Rn → R be a quadratic form with
associated matrix A and let λ1 , . . . , λn be the eigenvalues of A. Then

(1) Q is D+ if and only if λi > 0, for all i = 1, . . . , n.

(2) Q is D− if and only if λi < 0, for all i = 1, . . . , n.

(3) Q is SD+ if and only if λi ≥ 0, for all i = 1, . . . , n, and there is j ∈ {1, . . . , n} such
that λj = 0.

(4) Q is SD− if and only if λi ≤ 0, for all i = 1, . . . , n, and there is j ∈ {1, . . . , n}such
that λj = 0.

(5) Q is indefinite if and only if there are i, j ∈ {1, . . . , n}, such that λi > 0 and λj < 0.

2.1. Descartes’ Rule. Let p(x) = an xn + an−1 xn−1 + · · · + a1 x + a0 be a polynomial of a

single variable x, such that all its roots are real. Let

• r+ (p) be the number of positive roots of p, considering their multiplicity.

• r− (p) be the number of negative roots of p, considering their multiplicity.

• r0 (p) be the multiplicity of the null root.

• s(p) be the number of change of signs between the consecutive and non-null coeffi-
cients of p.

Then
r+ (p) = s(p), r− (p) = n − r+ (p) − r0 (p).

Remark 2.4. To classify a quadratic form Q, it suffices to know the sign of its eigenval-
ues, or roots of its characteristic polynomial. An important property of the characteristic
polynomial associated to a symmetric matrix is that all its roots are real (not true for an
arbitrary square matrix!). Thus, Descartes’ Rule applies, and allows us to classify easily
4 I.4 QUADRATIC FORMS

Q once we have calculated the characteristic polynomial, without knowing explicitly the
eigenvalues.

Example 2.5. Classify the quadratic form Q(x, y, z) = x2 + 2y 2 + z 2 + 2xz − 2yz.

The matrix associated to Q is

 
1 0 1
 
A=
 0 2 −1 

1 −1 1
and the characteristic polynomial pA (λ) = −λ3 +4λ2 −3λ−1. As pA comes from a symmetric
matrix, is has three real roots. As it is not homogeneous, r0 (pA ) = 0. There are two change
of signs in pA , thus s(pA ) = 2 and hence the polynomial has 2 positive roots and 3 − 2 = 1
negative root. In consequence, two eigenvalues are positive, and one is negative and the
quadratic from is indefinite.
 
a11 a12 ··· a1n
 
 a21 a22 · · · a2n 
Definition 2.6. Let A =  . .
 
 .. .. .. ..
 . . . 

an1 an2 · · · ann
The principal minor of order r of A is the determinant of the submatrix of A formed by the
first r rows and columns.
 
a11 a12 · · · a1r
 
 a21 a22 · · · a2r 
∆r = det  .
 
.. .. .. ..

 . . . . 

ar1 ar2 · · · arr

Theorem 2.7. Let Q : Rn → R be a quadratic form and let ∆1 , ∆2 , . . . , ∆n be the principal

minors of the associated matrix A. Then

• Q is D+ if and only if ∆1 > 0, ∆2 > 0,. . . ,∆n > 0.

• Q is D− if and only if ∆1 < 0, ∆2 > 0,. . . ,(−1)n ∆n > 0.

• If ∆n = det A 6= 0, and none of the two previous conditions are fulfilled, then Q is
indefinite.

• If ∆n = det A = 0 and rank A = p, then it is possible to exchange columns (and the

same rows at once), such that the principal minor of order p of the new matrix is
different from zero. Let ∆01 ,. . . ,∆0p be the principal minors of the new matrix. Then
 I.4 QUADRATIC FORMS 5

– Q is SD+ if and only if ∆01 > 0, ∆02 > 0,. . . ,∆0p > 0.

– Q is SD− if and only if ∆01 < 0, ∆02 > 0,. . . ,(−1)p ∆0p > 0.

– Q is indefinite if none of the two previous conditions hold true.

Remark 2.8. The exchange of columns and the corresponding rows in the matrix A is
equivalent to exchange the role of the corresponding variables.

Example 2.9. Classify the quadratic form Q(x, y, z) = x2 + 2y 2 + z 2 + 2xz − 2yz.

This quadratic form has been classified above as Indefinite with Descartes’ Rule. Of course,
we get the same result if we use the alternative principal minors method. The matrix
associated to Q is
 
1 0 1
 
A=
 0 2 −1 

1 −1 1

and ∆1 = 1 > 0, ∆2 = 2 > 0, ∆3 = |A| = −1 < 0.

3. Restricted quadratic forms

Let Q : Rn → R be a quadratic form and let V ⊆ Rn be given by

V = {x ∈ Rn : Bx = 0},

where B ∈ Mm×n , and rank B = m. Let Q V

: V → R the restriction of Q to V .

Definition 3.1. We say that Q restricted to V is D+ , D− , SD+ , SD− or indefinite, if and

only if Q V
is D+ , D− , SD+ , SD− or indefinite, respectively.

Remark 3.2. (1) If Q is D+ , then Q V

is D+ .

(2) If Q is D− , then Q V
is D− .

(3) If Q is SD+ , then Q V

may be SD+ or D+ .

(4) If Q is SD− , then Q V

may be SD− or D− .

(5) If Q is indefinite, then Q V

may be of any kind.
6 I.4 QUADRATIC FORMS

3.1. Classification criterion. Let Q : Rn → R be a quadratic form and let V ⊆ Rn be

given by
V = {x ∈ Rn : Bx = 0},

where B ∈ Mm×n , and rank B = m < n.

(1) By substitution. Solve Bx = 0 to obtain m variables that depend on n − m parame-

ters. By substituting into the expression of Q(x), the restricted quadratic form Q V
is obtained, which is an unrestricted quadratic form of n − m variables.

(2) By principal minors of the bordered matrix. Let the bordered matrix
!
O B
A∗ = .
Bt A

(a) If the last n − m principal minors of A∗ have the same sign as (−1)m , then Q V
is D+ .

(b) If the last n − m principal minors of A∗ alternate sign, starting with the sign
of (−1)m+1 , then Q V
is D− .

Example 3.3. Classify the quadratic form Q(x, y, z) = x2 + 2y 2 + z 2 + 2xz − 2yz restricted
to the subspace V defined by the equation 4x − 2y − z = 0 (the equation describes a plane).

The bordered matrix is  

0 4 −2 −1
 
 4 1
∗ 0 1 
A = .
 −2 0 2 −1
 

−1 1 −1 1
The subspace V is given by m = 1 equation and the number of variables is 3, thus we study
the last 3 − 1 = 2 principal minors of A∗ .

0 4 −2 −1
0 4 −2
4 1 0 1
4 1 0 = −36, = −54,
−2 0 2 −1
−2 0 2
−1 1 −1 1

both have the sign of (−1)m = (−1)1 = −1, thus the quadratic form restricted to V is D+ .

An alternative to this (computationally expensive) method is to plug z = 4x − 2y into Q,

so we get

q(x, y) = Q(x, y, 4x − 2y) = x2 + 2y 2 + (4x − 2y)2 + 2(4x − 2y)(x − y) = 25x2 + 10y 2 − 28xy,
 I.4 QUADRATIC FORMS 7
!
25 −14
with matrix , which is D+ .
−14 10