§1.22 Problem 85

Let A = delim{[}{
matrix{2}{2}{2 1 1 2}
}{]} and B = A^5 - 3 A^4 + 2A - I

(a) Determine the characteristic numbers and corresponding characteristic vectors of [B].

(b) Determine whether [B] is positive definite.

[B] = delim{[}{
matrix{2}{2}{122 121 121 122}
}{]}
- 3 delim{[}{
matrix{2}{2}{41 40 40 41}
}{]}
+2 delim{[}{
matrix{2}{2}{2 1 1 2}
}{]}
-delim{[}{
matrix{2}{2}{1 0 0 1}
}{]} = = delim{[}{
matrix{2}{2}{2 3 3 2}
}{]} .

|A - λI| = 0 = delim{|}{
matrix{2}{2}{(2-lambda) 3 3 (2-lambda)}
}{|} ⇒ (2-lambda)(2-lambda)-9 =0 ⇒ (2-lambda)^2=9 ⇒ (2-lambda)= pm 3 ⇒ lambda = 2 pm 3 ⇒

lambda_1 = 5 , lambda_2 = -1 .

[B] is not positive definite since a characteristic value is negative.

The elements of [A]¹⁰⁰ are on the order of 10⁴⁷.

n=101. r = (1,2,…,101)

$A^100 = P(A) = sum{k=1}{n}{P(lambda_k) Z_k (A)}$ .

$Z_1={A + I }/{6}$ , $Z_2={A - 5 I}/{-6}.$

$A^100 = {5^100} {{A + I }/{6}} - {-1^100}{A - 5 I}/{6}$ = $A^100 = {{5^100}/6} delim{[}{matrix{2}{2}{3 1 1 3}}{]} - {1/6} delim{[}{matrix{2}{2}{{-3} 1 1 {-3}}}{]}$ = $A^100 = {{5^100}/6} delim{[}{matrix{2}{2}{3 1 1 3}}{]} - {1/6} delim{[}{matrix{2}{2}{{-3} 1 1 {-3}}}{]}$

…which give elements on the order of 10⁶⁹, so this is clearly wrong, most likely because of some stupid arithmetic or algebra error.

§1.22 Problem 87

Suppose that the elements of a matrix [A](t) = [a_{ij}(t)] are differentiable functions of a variable t.

(a) From the definition
{dA(t)}/{dt} = lim{Delta t right 0}{{A(t+Delta t)-A(t)}/{Delta t}} ≡ lim{Delta t right 0}{ {Delta A}/{Delta t}} ,
prove that d[A](t)/dt = [da_ij/dt].

(b) Prove that {{d}/{dt}}(A B) = {{dA}/{dt}}B + A{{dB}/{dt}}

By definition, ${da_{ij}(t)}/{dt} = lim{Delta t right 0}{{a_{ij}(t+Delta t)-a_{ij}(t)}/{Delta t}} ≡ lim{Delta t right 0}{ {Delta a_{ij}}/{Delta t}}$

[Analogous to, “Prove that a chisel is most effective when pointed at a stone and struck on the blunt end.”]

§1.22 Problem 88

(a) If [A] is a real symmetric matrix, verify that the differential equation
{dx}/{dt} = Ax
is satisfied by x = e^tAc, where c is a constant vector.

(b) Use this result and an appropriate modification of equation (230) to solve the system
${dx_1}/dt = x_1 + 2 x_2,$
${dx_2}/dt = 2x_1 + x_2,$
subject to the initial conditions {x₁(0), x₂(0)} = {c₁, c₂}.