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Multivariate Distributions 795

The rank of a matrix. The rank of the n * m matrix A is the number of linearly independ-
ent columns of A. The rank of A is denoted rank(A). If the rank of A equals the number
of columns of A, then A is said to have full column rank. If the n * m matrix A has full
column rank, then there does not exist a nonzero m * 1 vector c such that Ac = 0n * 1. If
A is n * n with rank(A) = n, then A is nonsingular. If the n * m matrix A has full column
rank, then A′A is nonsingular.

The matrix square root. Let V be an n * n square symmetric positive definite matrix. The
matrix square root of V is defined to be an n * n matrix F such that F′F = V . The matrix
square root of a positive definite matrix will always exist, but it is not unique. The matrix
square root has the property that FV -1F′ = In. In addition, the matrix square root of a
positive definite matrix is invertible, so F′-1VF -1 = In.

Eigenvalues and eigenvectors. Let A be an n * n matrix. If the n * 1 vector q and the
scalar l satisfy Aq = lq, where q′q = 1, then l is an eigenvalue of A, and q is the eigen-
vector of A associated with that eigenvalue. An n * n matrix has n eigenvalues, which
need not take on distinct values, and n eigenvectors.

If V is an n * n symmetric positive definite matrix, then all the eigenvalues of V are
positive real numbers, and all the eigenvectors of V are real. Also, V can be written in
terms of its eigenvalues and eigenvectors as V = Q Q′, where is a diagonal n * n
matrix with diagonal elements that equal the eigenvalues of V, and Q is an n * n matrix
consisting of the eigenvectors of V, arranged so that the ith column of Q is the eigenvector
corresponding to the eigenvalue that is the ith diagonal element of . The eigenvectors are
orthonormal, so Q′Q = In.

Idempotent matrices. A matrix C is idempotent if C is square and CC = C. If C is an
n * n idempotent matrix that is also symmetric, then C is positive semidefinite and C
has r eigenvalues that equal 1 and n − r eigenvalues that equal 0, where r = rank(C)
(Exercise 18.10).

Appendix

18.2 Multivariate Distributions

This appendix collects various definitions and facts about distributions of vectors of ran-
dom variables. We start by defining the mean and covariance matrix of the n-dimensional
random variable V. Next we present the multivariate normal distribution. Finally, we sum-
marize some facts about the distributions of linear and quadratic functions of jointly nor-
mally distributed random variables.