Problem S2.2.7

(a) Find the eigenvalues and corresponding eigenvectors of unit length for the tensor T whose components in a certain coordinate system are:
row 1: (5/2), -(1/2), 0
row 2: -(1/2), (5/2), 0
row 3: 0, 0, 4

Work:

row 1: (5/2 - lamda), -1/2, 0
row 2: (-1/2, (5/2 - lamda), 0
row 3: 0, 0, (4 - lamda)
= 0
eigenvalues:
lamda^i, where i = 1, 2, 3
(5/2 - lamda)^2*(4 - lamda) = 0
lamda^(1) = lamda^(2) = 5/2, lamda^(3) = 4
eigenvectors:
lamda^(1), lamda^(2): v^(1)ae(1)
lamda^(3): v^(2)be(2)

(b) Verify that the eigenvectors are mutually orthogonal.

lamda^(1) = lamda^(2) = 5/2
v^(1) = ae^(1) and v^(2) = be^(2): v^(1)*v(2) = a*b = 0

(c) Give a geometric description of the rotation of axes that will put the coordinate matrix of T into diagonal form.

Must have non-zero numbers on the diagnonal of the matrix.

Solution

(a) eigenvalues: lamda^(1) = lamda^(2) = 5/2, lamda^(3) = 4

eigenvectors:
lamda^(1), lamda^(2): v^(1)ae(1)
lamda^(3): v^(2)be(2)

(b)lamda^(1) = lamda^(2) = 5/2
v^(1) = ae^(1) and v^(2) = be^(2): v^(1)*v(2) = a*b = 0

(c) Must have non-zero numbers on the diagnonal of the matrix.

Problem posted 08/03/05 -- LAL