3.2.2 Quadratic forms: conditions for definiteness
Definitions
Relevant 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?
 Definition

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).
 Example

x2
1 + x2
2 > 0 if (x_{1}, x_{2}) ≠ 0, so this quadratic form is positive definite. More generally, ax2
1 + cx2
2 is positive definite whenever a > 0 and c > 0
 Example

x2
1 + 2x_{1}x_{2} + x2
2 may be expressed as (x_{1} + x_{2})^{2}, which is nonnegative for all (x_{1}, x_{2}). Thus this quadratic form is positive semidefinite. It is not positive definite because (x_{1} + x_{2})^{2} = 0 for (x_{1}, x_{2}) = (1,−1) (for example).
 Example

x2
1 − x2
2 > 0 for (x_{1}, x_{2}) = (1, 0) (for example), and x2
1 − x2
2 < 0 for (x_{1}, x_{2}) = (0, 1) (for example). Thus this quadratic form is indefinite.
Two variables
We 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 x_{1} and x_{2}. Consider the quadratic formGiven a ≠ 0, we have
Q(x, y)  =  a[(x + (b/a)y)^{2} + (c/a − (b/a)^{2})y^{2}]. 
Now, we have Q(1, 0) = a and Q(−b/a, 1) = (ac − b^{2})/a. Thus, if Q(x, y) is positive definite then a > 0 and ac > b^{2}.
We conclude that Q(x, y) is positive definite if and only if a > 0 and ac > b^{2}.
A similar argument shows that Q(x, y) is negative definite if and only if a < 0 and ac > b^{2}.
Note that if a > 0 and ac > b^{2} then because b^{2} ≥ 0 for all b, we can conclude that c > 0. Similarly, if a < 0 and ac > b^{2} 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 − b^{2}, 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 − b^{2} is the determinant of the matrix that represents the quadratic form, namely
A = 

. 
 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 variables
To obtain conditions for an nvariable quadratic form to be positive or negative definite, we need to examine the determinants of some of its submatrices. Definition

The kth order leading principal minor of the n × n symmetric matrix A = (a_{ij}) is the determinant of the matrix obtained by deleting the last n − k rows and columns of A (where k = 1, ... , n):
D_{k} = a_{11} a_{12} ... a_{1k} a_{21} a_{22} ... a_{2k} ... ... ... ... a_{k1} a_{k2} ... a_{kk} .
 Example

Let
A = 3 1 2 1 −1 3 2 3 2 . D_{2} = 3 1 1 −1 ,
 Proposition

Let A be an n × n symmetric matrix and let D_{k} for k = 1, ... , n be its leading principal minors. Then
 A is positive definite if and only if D_{k} > 0 for k = 1, ..., n.
 A is negative definite if and only if (−1)^{k}D_{k} > 0 for k = 1, ..., n. (That is, if and only if the leading principal minors alternate in sign, starting with negative for D_{1}.)
 Source
 For proofs of the first point, see Simon and Blume (1994), Theorem 16.1 (p. 394) Heal, Hughes, and Tarling (1974), T.55 (p. 106), and Hadley (1961), pp. 260–261. The argument for the second point is similar.
A = 

 Example

Let
A = −3 2 0 2 −3 0 0 0 −5 .
 Example

We saw above that the leading principal minors of the matrix
A = 3 1 2 1 −1 3 2 3 2