Linear Algebra, Theory And Applications, 2012a

3.6. EXERCISES 103
derivatives so it makes sense to write Ly. Suppose Ly k =0fork =1, 2, ··· ,n. The
Wronskian of these functions, y i is defined as
⎛
W (y 1 , ··· ,y n )(x) ≡ det ⎜
⎝
y 1 (x) ··· y n (x)
y 1 ′ (x) ··· y n ′ (x)
.
.
1 (x) ··· y n
(n−1)
y (n−1)
(x)
⎞
⎟
⎠
Show that for W (x) =W (y 1 , ··· ,y n )(x) to save space,
⎛
W ′ (x) =det⎜
⎝
y 1 (x) ··· y n (x)
. ···
.
1 (x) y n
(n−2) (x)
y (n)
1 (x) ··· y n
(n) (x)
y (n−2)
⎞
⎟
⎠ .
Now use the differential equation, Ly = 0 which is satisfied by each of these functions,
y i and properties of determinants presented above to verify that W ′ +a n−1 (x) W =0.
Give an explicit solution of this linear differential equation, Abel’s formula, and use
your answer to verify that the Wronskian of these solutions to the equation, Ly =0
either vanishes identically on (a, b) or never.
8. Two n × n matrices, A and B, are similar if B = S −1 AS for some invertible n × n
matrix S. Show that if two matrices are similar, they have the same characteristic
polynomials. The characteristic polynomial of A is det (λI − A) .
9. Suppose the characteristic polynomial of an n × n matrix A is of the form
t n + a n−1 t n−1 + ···+ a 1 t + a 0
and that a 0 ≠0. Find a formula A −1 intermsofpowersofthematrixA. Show that
A −1 exists if and only if a 0 ≠0. In fact, show that a 0 =(−1) n det (A) .
10. ↑Letting p (t) denote the characteristic polynomial of A, show that p ε (t) ≡ p (t − ε)
is the characteristic polynomial of A + εI. Then show that if det (A) =0, it follows
that det (A + εI) ≠ 0 whenever |ε| is sufficiently small.
11. In constitutive modeling of the stress and strain tensors, one sometimes considers sums
of the form ∑ ∞
k=0 a kA k where A is a 3×3 matrix. Show using the Cayley Hamilton
theorem that if such a thing makes any sense, you can always obtain it as a finite sum
havingnomorethann terms.
12. Recall you can find the determinant from expanding along the j th column.
det (A) = ∑ i
A ij (cof (A)) ij
Think of det (A) as a function of the entries, A ij . Explain why the ij th cofactor is
really just
∂ det (A)
∂A ij
.