DefinitionsRelevant questions when we use quadratic forms in studying the concavity and convexity of functions of many variables are:
- Under what condition on the matrix A are the values of the quadratic form Q(x) = x'Ax positive for all values of x ≠ 0?
- Under what condition are these values negative for all values of x ≠ 0?
Let Q be a quadratic form, and let A be the symmetric matrix that represents it (i.e. Q(x) = x'Ax for all x). Then Q (and the associated matrix A) is
- positive definite if x'Ax > 0 for all x ≠ 0
- negative definite if x'Ax < 0 for all x ≠ 0
- positive semidefinite if x'Ax ≥ 0 for all x
- negative semidefinite if x'Ax ≤ 0 for all x
- indefinite if it is neither positive nor negative semidefinite (i.e. if x'Ax > 0 for some x and x'Ax < 0 for some x).
1 + x2
2 > 0 if (x1, x2) ≠ 0, so this quadratic form is positive definite. More generally, ax2
1 + cx2
2 is positive definite whenever a > 0 and c > 0
1 + 2x1x2 + x2
2 may be expressed as (x1 + x2)2, which is nonnegative for all (x1, x2). Thus this quadratic form is positive semidefinite. It is not positive definite because (x1 + x2)2 = 0 for (x1, x2) = (1,−1) (for example).
1 − x2
2 > 0 for (x1, x2) = (1, 0) (for example), and x2
1 − x2
2 < 0 for (x1, x2) = (0, 1) (for example). Thus this quadratic form is indefinite.
Two variablesWe can easily derive conditions for the definiteness of any quadratic form in two variables. To make the argument more readable, I change the notation slightly, using x and y for the variables, rather than x1 and x2. Consider the quadratic form
Given a ≠ 0, we have
|Q(x, y)||=||a[(x + (b/a)y)2 + (c/a − (b/a)2)y2].|
Now, we have Q(1, 0) = a and Q(−b/a, 1) = (ac − b2)/a. Thus, if Q(x, y) is positive definite then a > 0 and ac > b2.
We conclude that Q(x, y) is positive definite if and only if a > 0 and ac > b2.
A similar argument shows that Q(x, y) is negative definite if and only if a < 0 and ac > b2.
Note that if a > 0 and ac > b2 then because b2 ≥ 0 for all b, we can conclude that c > 0. Similarly, if a < 0 and ac > b2 then c < 0. Thus, to determine whether a quadratic form is positive or negative definite we need to look only at the signs of a and of ac − b2, but if the conditions for positive definiteness are satisfied then it must in fact also be true that c > 0, and if the conditions for negative definitely are satisfied then we must also have c < 0.
Notice that ac − b2 is the determinant of the matrix that represents the quadratic form, namely
- positive definite if and only if a > 0 and |A| > 0 (in which case c > 0)
- negative definite if and only if a < 0 and |A| > 0 (in which case c < 0)
Many variablesTo obtain conditions for an n-variable quadratic form to be positive or negative definite, we need to examine the determinants of some of its submatrices.
The kth order leading principal minor of the n × n symmetric matrix A = (aij) is the determinant of the matrix obtained by deleting the last n − k rows and columns of A (where k = 1, ... , n):
Dk = a11 a12 ... a1k a21 a22 ... a2k ... ... ... ... ak1 ak2 ... akk .
A = 3 1 2 1 −1 3 2 3 2 . D2 = 3 1 1 −1 ,
- Let A be an n × n symmetric matrix and let Dk for k = 1, ... , n be its leading principal minors. Then
A = −3 2 0 2 −3 0 0 0 −5 .
We saw above that the leading principal minors of the matrix
A = 3 1 2 1 −1 3 2 3 2