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.