3.2.3 Quadratic forms: conditions for semidefiniteness
Two variables
First consider the case of a twovariable quadratic formNow assume a ≠ 0. As before, we have
Q(x, y)  =  a[(x + (b/a)y)^{2} + (c/a − (b/a)^{2})y^{2}]. 
We conclude that if a ≥ 0, c ≥ 0, and ac − b^{2} ≥ 0, then the quadratic form is positive semidefinite.
Conversely, if the quadratic form is positive semidefinite then Q(1, 0) = a ≥ 0, Q(0, 1) = c ≥ 0, and Q(−b, a) = a(ac − b^{2}) ≥ 0. If a = 0 then by the previous argument we need b = 0 and c ≥ 0 in order for the quadratic form to be positive semidefinite, so that ac − b^{2} = 0; if a > 0 then we need ac − b^{2} ≥ 0 in order for a(ac − b^{2}) ≥ 0.
We conclude that the quadratic form is positive semidefinite if and only if a ≥ 0, c ≥ 0, and ac − b^{2} ≥ 0.
A similar argument implies that the quadratic form is negative semidefinite if and only if a ≤ 0, c ≤ 0, and ac − b^{2} ≥ 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 ≥ 0 and ac − b^{2} ≥ 0, it is not necessarily the case that c ≥ 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 − b^{2} ≥ 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) = ax^{2} + 2bxy + cy^{2} is
 positive semidefinite if and only if a ≥ 0, c ≥ 0, and A ≥ 0
 negative semidefinite if and only if a ≤ 0, c ≤ 0, and A ≥ 0
A = 

. 
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 .
 Example

Let
A = 3 1 2 1 −1 3 2 3 2 .  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
 the last row and last column
 the second row and second column
 the first row and first column
 Proposition

Let A be an n × n symmetric matrix. Then
 A is positive semidefinite if and only if all its principal minors are nonnegative.
 A is negative semidefinite if and only if its kth order principal minors are nonpositive for k odd and nonnegative for k even.
 Source
 For a proof, see Gantmacher (1959), Theorem 4 on p. 307.
 Example

Let
A = 0 0 0 −1 .
Procedure for checking the definiteness of a matrix
 Procedure for checking the definiteness of a matrix

 Find the leading principal minors and check if the conditions for positive or negative definiteness are satisfied. If they are, you are done. (If a matrix is positive definite, it is certainly positive semidefinite, and if it is negative definite, it is certainly negative semidefinite.)
 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 D_{1} = 1, D_{2} = 0, and D_{3} = −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 (D_{1} is positive while D_{3} is negative), so the matrix is indefinite.
 Example
 Suppose that the leading principal minors of the 3 × 3 matrix A are D_{1} = 1, D_{2} = 0, and D_{3} = 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.