Page 57 - Introduction to Continuum Mechanics

42 Tensors

Thus (note the second and third equations are the same),

and the eigenvector corresponding to A2=5 is

Corresponding to ^3= -5, similar computations give

All the examples given above have three eigenvalues that are real. It can be shown that if a
tensor is real (i.e., with real components) and symmetric, then all its eigenvalues are real. If a
tensor is real but not symmetric, then two of the eigenvalues may be complex conjugates. The
following example illustrates this possibility.

Example 2B17.5

Find the eigenvalues and eigenvectors for the rotation tensor R corresponding to a 90°
rotation about the e 3-axis (see Example 2B5.1(a)).

Solution. The characteristic equation is

I.e.,

V / \ / \~ S\ - f -
±
Thus, only one eigenvalue is real, namely Aj-1, the other two are imaginary, &2,3- ^~~l*
Correspondingly, there is only one real eigenvector. Only real eigenvectors are of interest to
us, we shall therefore compute only the eigenvector corresponding to Aj=1.
From

and