Page 248 - Probability and Statistical Inference

4. Functions of Random Variables and Sampling Distribution 225

A square matrix A n×n is called orthogonal if and only if A’, the transpose of
A, is the inverse of A. If A n×n is orthogonal then it can be checked that det(A)
= ±1.
A square matrix A = (a ) is called symmetric if and only if A’ = A, that
ij n×n
is a = a , 1 ≤ i ≠ j ≤ n.
ji
ij
Let A = (a , i,j = 1, ..., n, be a n×n matrix. Denote the l×l sub-matrix B =
ij
l
th
(a ), p, q = 1, ..., l,l = 1, ..., n. The l principal minor is defined as the
pq
det(B ), l = 1, ..., n.
l
For a symmetric matrix A , an expression such as x’ Ax with x ∈ ℜ is
n
n×n
customarily called a quadratic form.
A symmetric matrix A n×n is called positive semi definite (p.s.d.) if (a) the
quadratic form x’ Ax ≥ 0 for all x ∈ ℜ , and (b) the quadratic form x’ Ax =0
k
for some non-zero x ∈ ℜ .
n
A symmetric matrix A is called positive definite (p.d.) if (a) the qua-
n×n
n
dratic form x’ Ax ≥ 0 for all x ∈ ℜ , and (b) the quadratic form x’ Ax =0 if
and only if x = 0.
A symmetric matrix A n×n is called negative definite (n.d.) if (a) the qua-
n
dratic form x’ Ax ≤0 for all x ∈ ℜ , and (b) the quadratic form x’ Ax =0 if and
only if x = 0. In other words, a symmetric matrix A n×n is n.d. if and only if –
A is positive definite.

Let us now look at some partitioned matrices A, B where

where m, n, u, v, w, t, c, d are all positive integers such that u + w = m, u + t
= n and c + d = q. Then, one has