A72 Idempotent Quadratic Forms
Quadratic forms in idempotent matrices play an important role in the distributions of many test statistics. As such, we shall encounter them fairly often. Two central results are of interest.
• Every symmetric idempotent matrix is nonnegative definite. (A-116)
Proof: All roots are one or zero; hence, the matrix is nonnegative definite by definition.
Combining this with some earlier results yields a result used in determining the sampling distribution of most of the standard test statistics.
• If A is symmetric and idempotent, n x n with rank /, then every quadratic form in A can be written x'Ax = ]Ty=1 y] (A-117)
Proof: This result is (A-110) with X = one or zero. A.7.3 COMPARING MATRICES
Derivations in econometrics often focus on whether one matrix is "larger" than another. We now consider how to make such a comparison. As a starting point, the two matrices must have the same dimensions. A useful comparison is based on d = x'Ax - x'Bx = x'(A - B)x.
If d is always positive for any nonzero vector, x, then by this criterion, we can say that A is larger than B. The reverse would apply if d is always negative. It follows from the definition that if d > 0 for all nonzero x, then A — B is positive definite. (A-118)
If d is only greater than or equal to zero, then A — B is nonnegative definite. The ordering is not complete. For some pairs of matrices, d could have either sign, depending on x. In this case, there is no simple comparison.
A particular case of the general result which we will encounter frequently is:
If A is positive definite and B is nonnegative definite, then A + B > A. (A-119)
Consider, for example, the "updating formula" introduced in (A-66). This uses a matrix
Finally, in comparing matrices, it may be more convenient to compare their inverses. The result analogous to a familiar result for scalars is:
In order to establish this intuitive result, we would make use of the following, which is proved in Goldberger (1964, Chapter 2):
THEOREM A.. 12 Ordering for Positive Definite Matrices
If A and B are two positive definite matrices with the same dimensions and if every characteristic root of A is larger than (at least as large as) the corresponding characteristic root of B when both sets of roots are ordered from largest to smallest, then A — B is positive (nonnegative) definite.
The roots of the inverse are the reciprocals of the roots of the original matrix, so the theorem can be applied to the inverse matrices.
.8 CALCULUS AND MATRIX ALGEBRA14
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