Matrices and Tensors - Continuum Mechanics of Anisotropic Materials - page 394

X

k¼n

ð A B Þ ij ¼ A ik B kj ; (A.20)

k¼ 1

thus, for example, the element in the r th row and c th column of the product A B is

given by

ð A B Þ rc ¼ A r1 B 1c þ A r2 B 2c þþ A rn B nc :

The widely used notational convention, called the Einstein summation conven-
tion , allows one to simplify the notation by dropping the summation symbol in
(A.20) so that

ð A B Þ ij ¼ A ik B kj ;

(A.21)

where the convention is the understanding that the repeated index, in this case k ,is
to be summed over its range of the admissible values from 1 to n . For n 6, the

range of admissible values is 1-6, including 2, 3, 4, and 5. The two k indices are the
summation or dummy indices; note that the implied summation is unchanged if both
of the k ' s are replaced by any other letter of the alphabet.

¼

The Summation Index and Summands

A summation index is defined as an index that occurs in a summand twice and only
twice. Note that summands are terms in equations separated from each other by
plus, minus, or equal signs. The existence of summation indices in a summand
requires that the summand be summed with respect to those indices over the entire
range of admissible values. Note again that the summation index is only a means of
stating that a term must be summed, and the letter used for this index is immaterial,
thus A im B mj has the same meaning as A ik B kj or A k B k.

The Free Index

The other indices in the formula (A.21), the i and j indices, are called free indices.
A free index is free to take on any one of the admissible values in its range from1 to n .
For example if n were 3, the free index could be 1, 2, or 3. A free index is formally
defined as an index that occurs once and only once in every summand of an equation.
The total number of equations that may be represented by an equation with one free
index is the range of the admissible values. Thus the equation (A.21) represents
n 2 separate equations. For two 2 by 2 matrices A and B , the product is written as