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794 Chapter 18 The Theory of Multiple Regression

that A has dimension n * m and B has dimension m * r. Then the product of A and B is
m
an n * r matrix, C; that is, C = AB, where the (i, j) element of C is cij = g k= 1aikbkj. Said

differently, the (i, j) element of AB is the product of multiplying the row vector that is the

ith row of A with the column vector that is the j th column of B.

The product of a scalar d with the matrix A has the (i, j) element daij; that is, each
element of A is multiplied by the scalar d.

Some useful properties of matrix addition and multiplication. Let A and B be matrices.
Then:

a. A + B = B + A;
b. (A + B) + C = A + (B + C);
c. (A + B)′ = A′ + B′;
d. If A is n * m, then AIm = A and InA = A;
e. A(BC) = (AB)C;
f. (A + B)C = AC + BC; and
g. (AB)′ = B′A′.

In general, matrix multiplication does not commute; that is, in general AB BA,
although there are some special cases in which matrix multiplication commutes; for exam-
ple, if A and B are both n * n diagonal matrices, then AB = BA.

Matrix Inverse, Matrix Square Roots, and Related Topics

The matrix inverse. Let A be a square matrix. Assuming that it exists, the inverse of the
matrix A is defined as the matrix for which A−1A = In. If in fact the inverse matrix A−1
exists, then A is said to be invertible or nonsingular. If both A and B are invertible, then
(AB)−1 = B−1A−1.

Positive definite and positive semidefinite matrices. Let V be an n * n square matrix.
Then V is positive definite if c′Vc 7 0 for all nonzero n * 1 vectors c. Similarly, V is
positive semidefinite if c′Vc Ú 0 for all nonzero n * 1 vectors c. If V is positive definite,
then it is invertible.

Linear independence. The n * 1 vectors a1 and a2 are linearly independent if there do not exist
nonzero scalars c1 and c2 such that c1a1 + c2a2 = 0n * 1. More generally, the set of k vectors
a1, a2, c, ak are linearly independent if there do not exist nonzero scalars c1, c2, c, ck such
that c1a1 + c2a2 + g + ckak = 0n * 1.