Linear Algebra, Theory And Applications, 2012a

76 MATRICES AND LINEAR TRANSFORMATIONS
38. In the context of Problem 37, show there exists a nonzero vector x such that
(A − λ j I) x = 0.
This is called an eigenvector and the λ j is called an eigenvalue. Hint:There must exist
a vector y such that
(A − λ 1 I) r1 (A − λ 2 I) r2 ···(A − λ j I) rj−1 ···(A − λ k I) r k
y = z ≠ 0
Why? Now what happens if you do (A − λ j I)toz?
39. Suppose Q (t) is an orthogonal matrix. This means Q (t) isarealn × n matrix which
satisfies
Q (t) Q (t) T = I
Suppose also the entries of Q (t) are differentiable. Show ( Q T ) ′
= −Q T Q ′ Q T .
40. Remember the Coriolis force was 2Ω × v B where Ω was a particular vector which
came from the matrix Q (t) as described above. Show that
⎛
i (t) · i (t 0 ) j (t) · i (t 0 ) k (t) · i (t 0 )
⎞
Q (t) = ⎝ i (t) · j (t 0 ) j (t) · j (t 0 ) k (t) · j (t 0 ) ⎠ .
i (t) · k (t 0 ) j (t) · k (t 0 ) k (t) · k (t 0 )
There will be no Coriolis force exactly when Ω = 0 which corresponds to Q ′ (t) =0.
When will Q ′ (t) =0?
41. An illustration used in many beginning physics books is that of firing a rifle horizontally
and dropping an identical bullet from the same height above the perfectly
flat ground followed by an assertion that the two bullets will hit the ground at exactly
the same time. Is this true on the rotating earth assuming the experiment
takes place over a large perfectly flat field so the curvature of the earth is not an
issue? Explain. What other irregularities will occur? Recall the Coriolis acceleration
is 2ω [(−y ′ cos φ) i+(x ′ cos φ + z ′ sin φ) j − (y ′ sin φ) k] wherek points away from the
center of the earth, j points East, and i points South.