Page 186 - Intro to Tensor Calculus

181

Therefore, associated with distinct eigenvalues λ (i) ,i =1, 2, 3 there are unit eigenvectors

ˆ
L (i) = ` i1 ˆ e 1 + ` i2 ˆ e 2 + ` i3 ˆ e 3

with components ` im ,m =1, 2, 3 which are direction cosines and satisfy

` in ` im = δ mn and ` ij ` jm = δ im . (2.1.23)

The unit eigenvectors satisfy the relations

T ij ` j1 = λ (1) ` i1 T ij ` j2 = λ (2) ` i2 T ij ` j3 = λ (3) ` i3

and can be written as the single equation

T ij ` jm = λ (m) ` im , m =1, 2,or 3 m not summed.

Consider the transformation

x i = ` ij x j or x m = ` mj x j

which represents a rotation of axes, where ` ij are the direction cosines from the eigenvectors of T ij . This is a
linear transformation where the ` ij satisfy equation (2.1.23). Such a transformation is called an orthogonal
transformation. In the new x coordinate system, called principal axes, we have

i
∂x ∂x j
T mn = T ij m n = T ij ` im ` jn = λ (n) ` in ` im = λ (n) δ mn (no sum on n). (2.1.24)
∂x ∂x
This equation shows that in the barred coordinate system there are the components
 
0 0
λ (1)

T mn =  0 λ (2) 0  .
0 0 λ (3)
That is, along the principal axes the tensor components T ij are transformed to the components T ij where
T ij =0for i 6= j. The elements T (i)(i) , i not summed, represent the eigenvalues of the transformation
(2.1.19).