Quadratic forms: conditions for semidefiniteness

Two variables

First consider the case of a two-variable quadratic form

Q(x, y) = ax² + 2bxy + cy².

If a = 0 then Q(x, 1) = 2bx + c, which is nonnegative for all values of x if and only if b = 0 and c ³ 0, in which case ac - b² = 0, and is nonpositive for all values of x if and only if b = 0 and c £ 0, in which case ac - b² = 0. Now consider the case in which a ¹ 0. As before, we can write

Q(x, y) = a[(x + (b/a)y)² + (c/a - (b/a)²)y²].

When is this expression nonnegative? Both squares are nonnegative, so we need a ³ 0 and ac - b² ³ 0. Similarly, the expression is nonpositive if a £ 0 and ac - b² ³ 0. We conclude that if a ³ 0, c ³ 0, and ac - b² ³ 0, then the quadratic form is positive semidefinite, and if a £ 0, c £ 0, and ac - b² ³ 0 then it is negative semidefinite. Conversely, if the quadratic form is positive semidefinite then Q(1, 0) = a ³ 0, Q(0, 1) = c ³ 0, and Q(-b, a) = a(ac - b²) ³ 0. If a = 0 then by the previous argument we need b = 0 and c ³ 0, so that ac - b² = 0; if a > 0 then we need ac - b² ³ 0 in order for ac - b² ³ 0. Symmetric arguments apply to the case of negative semidefiniteness. We conclude that the quadratic form is positive semidefinite if and only if a ³ 0, c ³ 0, and ac - b² ³ 0, and negative semidefinite if and only if a £ 0, c £ 0, and ac - b² ³ 0.

Note that in this case, unlike the case of positive and negative definiteness, we need to check all three conditions, not just two of them. If a ³ 0 and ac - b² ³ 0, it is not necessarily the case that c ³ 0 (try a = b = 0 and c < 0), so that the quadratic form is not necessarily positive semidefinite. (Similarly, the conditions a £ 0 and ac - b² ³ 0 are not sufficient for the quadratic form to be negative semidefinite: we need, in addition, c £ 0.)

Thus we can rewrite the results as follows: the two variable quadratic form Q(x, y) = ax² + 2bxy + cy² is

positive semidefinite if and only if a ³ 0, c ³ 0, and ½A½ ³ 0

negative semidefinite if and only if a £ 0, c £ 0, and ½A½ ³ 0

where

A =

a b

b c

It follows that the quadratic form is indefinite if and only if ½A½ < 0. (Note that if ½A½ ³ 0 then ac ³ 0, so we cannot have a < 0 and c ³ 0, or a ³ 0 and c £ 0.)

Many variables

As in the case of two variables, to determine whether a quadratic form is positive or negative semidefinite we need to check more conditions than we do in order to check whether it is positive or negative definite. In particular, it is not true that a quadratic form is positive or negative semidefinite if the inequalities in the conditions for positive or negative definiteness are satisfied weakly. In order to determine whether a quadratic form is positive or negative semidefinite we need to look at more than simply the leading principal minors. The matrices we need to examine are described in the following definition.

Definition
The kth order principal minors of an n ´ n symmetric matrix A are the determinants of the k ´ k matrices obtained by deleting n - k rows and the corresponding n - k columns of A (where k = 1, ... , n).

Note that the kth order leading principal minor of a matrix is one of its kth order principal minors.

Example
Let
A =

a b

b c

The first-order principal minors of A are a and c, and the second-order principal minor is the determinant of A, namely ac - b².

Example
Let
A =

3 1 2

1 -1 3

2 3 2

This matrix has 3 first-order principal minors, obtained by deleting

the last two rows and last two columns
the first and third rows and the first and third columns
the first two rows and first two columns
which gives us simply the elements on the main diagonal of the matrix: 3, -1, and 2. The matrix also has 3 second-order principal minors, obtained by deleting

the last row and last column
the second row and second column
the first row and first column
which gives us -4, 2, and -11. Finally, the matrix has one third-order principal minor, namely its determinant, -19.

The following result gives criteria for semidefiniteness.
Proposition
Let A be an n ´ n symmetric matrix. Then

A is positive semidefinite if and only if all the principal minors of A are nonnegative.
A is negative semidefinite if and only if all the kth order principal minors of A are £ 0 if k is odd and ³ 0 if k is even.

Example
Let
A =

0 0

0 -1

The two first-order principal minors and 0 and -1, and the second-order principal minor is 0. Thus the matrix is negative semidefinite. (It is not negative definite, since the first (and the second) leading principal minor is zero.)

Recipe for checking definiteness of a matrix

Find the leading principal minors and check if conditions for positive or negative definiteness are satisfied.
If the conditions are not satisfied, check if they are strictly violated. If they are, then the matrix is indefinite.
If the conditions are not strictly violated, find all its principal minors and check if the conditions for positive or negative semidefiniteness are satisfied.

Example: Suppose that the leading principal minors of the 3 ´ 3 matrix A are D₁ = 1, D₂ = 0, and D₃ = -1. Neither the conditions for A to be positive definite nor those for A to be negative definite are satisfied. In fact, both conditions are strictly violated (D₁ is positive while D₃ is negative), so the matrix is indefinite.

Example
Suppose that the leading principal minors of the 3 ´ 3 matrix A are D₁ = 1, D₂ = 0, and D₃ = 0. Neither the conditions for A to be positive definite nor those for A to be negative definite are satisfied. But the condition for positive definiteness is not strictly violated. To check semidefiniteness, we need to examine all the principal minors.

Quadratic forms and quadratic functions

You may be wondering: why look only at quadratic forms?---why not consider arbitrary quadratic functions? In the case of two variables, such a function takes the form

f (x, y) = ax² + 2bxy + cy² + px + qy + r.

By studying quadratic forms we also study quadratic functions: by changing the variables we can reduce any quadratic function to a quadratic form. In the case of two variables, choose x, h, and d so that

2ax + 2bh = p, 2bx + 2ch = q, and ax² + 2bxh + ch² + d = r.

Then the quadratic function above can be written as

f (x, y) = a(x + x)² + 2b(x + x)(y + h) + c(y + h)² + d,

which, ignoring the constant d, is a quadratic form in the variables x + x and y + h. The only problem is that it might not be possible to choose x, h, and d to satisfy the three conditions. It turns out that it is possible if ac ¹ b²; if ac = b² then f (x, y) can be expressed differently as a simple quadratic function. (See pp. 529--530 in the text for details.)

Exercises