Matrices and Tensors - Continuum Mechanics of Anisotropic Materials - page 335

tr ð dev A Þ¼ tr A 1 = n ð tr A Þð tr 1 Þ;

then, since tr 1

¼ n , it follows that tr(dev A ) ¼

0.
The product of two square matrices, A and B , with equal numbers of rows
(columns) is a square matrix with the same number of rows (columns). The matrix
product is written as AB where AB is defined by

X

k ¼ n

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 AB 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 . This summation
convention will be used from this point forward in this Appendix and in the body of
the text. 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. 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 . The other indices in the
formula (A.22), 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 from 1 to n . For example if n
were3,thefreeindexcouldbe1,2,or3.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

¼