Matrices and Tensors - Continuum Mechanics of Anisotropic Materials - page 355

The components of the tensor T relative to the Latin basis, T (L) ¼ [ T ij ], are

related to the components relative to the Greek basis, T (G) ¼ [ T ab ], by

T ð L Þ ¼ T ð G Þ Q T and T ð G Þ ¼ Q T T ð L Þ Q Q : (A.83)

These formulas relating the components are the tensorial equivalent of vectorial
formulas v ð L Þ ¼ Q T

v ð L Þ given by (A.77), and their derivation is
similar. First, substitute the second of (A.66) into the (A.80) twice, once for each
base vector:

v ð G Þ and v ð G Þ ¼

Q

T ¼ T ij e i e j ¼ T ab Q i a Q j b e i e j : (A.84)

Then gather together the terms referred to the basis e i e j , thus

ð T ij T ab Q i a Q j b Þ e i e j ¼ 0 : (A.85)

Next take the scalar product of (A.85), first with respect to e k , and then with
respect to e m . One finds that the only nonzero terms that remain are

T km ¼ Q k a T ab Q m b : (A.86)

A comparison of the last term in (A.86) with the definition of matrix product
(A.20) suggests that it is a triple matrix product involving Q twice and T (G) once.
Careful comparison of the last term in (A.86) with the definition of matrix product
(A.20) shows that the summation is over a different index in the third element of the
product. In order for the last term in (A.86) to represent a triple matrix product, the
b index should appear as the first subscripted index rather than the second. How-

ever, this index may be relocated in the second matrix by using the transposition

operation as shown in the first equation of (A.21). Thus the last term in equation
(A.86) is the matrix product of Q · T with Q T . The result is the first equation of
(A.83). If the first, rather than the second, of (A.67) is substituted into the second
equality of (A.80), and similar algebraic manipulations accomplished, one obtains
the second equation of (A.83).
The word tensor is used to refer to the quantity T defined by (A.80), a quantity
independent of any basis. It is also used to refer to the matrix of tensor components
relative to a particular basis, for example T (L)

b

[ T ij ]or T (G) [ T ab ]. In both cases

“tensor” should be “tensor of order two,” but the order of the tensor is generally
clear from the context. A tensor of order N in a space of n dimensions is defined by

¼ ¼

B ¼ B ij ... k e i e j ... e k ¼ B ab...g e a e b ... e g : (A.87)

The number of base vectors in the basis is the order N of the tensor. It is easy to see
that this definition specializes to that of the second order tensor (A.80). The definition
of a vector as a tensor of order one is easy to see, and the definition of a scalar as a
tensor of order 0 is trivial.