Introduction to the Mathematical and Statistical Foundations of Econometrics
Positive Definite and Semidefinite Matrices
Another set of corollaries of Theorem I.36 concern positive (semi)definite matrices. Most of the symmetric matrices we will encounter in econometrics are positive (semi)definite or negative (semi)definite. Therefore, the following results are of the utmost importance to econometrics.
Definition I.23: Ann x n matrix A is called positive definite if, for arbitrary vectors x є К” unequal to the zero vector, xT Ax > 0, and it is called positive semidefinite if for such vectors x, xTAx > 0. Moreover, A is called negative (semi)definite if —A is positive (semi)definite.
Note that symmetry is not required for positive (semi)definiteness. However, x TAx can always be written as
x T Ax = x T^-A + - AT^ x = x T Asx, (I.62)
for example, where As is symmetric; thus, A is positive or negative (semi)definite if and only if As is positive or negative (semi)definite.
Theorem I.39: A symmetric matrix is positive (semi)definite if and only if all its eigenvalues are positive (nonnegative).
Proof: This result follows easily from xTAx = xT QЛ QTx = yTAy = J2j к jyj, where y = QTx with components yj. Q. E.D.
On the basis of Theorem I.39, we can now define arbitrary powers of positive definite matrices:
Definition I.24: If A is a symmetric positive (semi)definite n x n matrix, then for а є К [a > 0] the matrix A to the power a is defined by Aa = QЛa QT, where Ла is a diagonal matrix with diagonal elements the eigenvalues of A to the power a : Ла = diag(X<a,.X<a) and Q is the orthogonal matrix of corresponding eigenvectors.
The following theorem is related to Theorem I.8.
Theorem I.40: If A is symmetric and positive semidefinite, then the Gaussian elimination can be conducted without need for row exchanges. Consequently, there exists a lower-triangular matrix L with diagonal elements all equal to 1 and a diagonal matrix D such that A = LDLT.
Proof: First note that by Definition 1.24 with a = 1/2, A1/2 is symmetric and (A1/2)tA1/2 = A1/2A1/2 = A. Second, recall that, according to Theorem I.17 there exists an orthogonal matrix Q and an upper-triangular matrix U such that A1/2 = QU; hence, A = (A1/2)TA1/2 = UTQTQU = UTU. The matrix UT is lower triangular and can be written as UT = LD„, where D„ is a diagonal matrix and L is a lower-triangular matrix with diagonal elements all equal to 1. Thus, A = LDkD„LT = LDLT, where D = D„D„. Q. E.D.
