Ax for any x ∈ Rn

1. hw 9

(1) Let L(Rⁿ, R^m) be the space of all linear maps from Rⁿ to R^m. For each T ∈ L(Rⁿ, R^m), we define

kT k_op= sup

kxk_Rn=1

kT (x)k_Rⁿ. (a) Prove that k · k_op defines a norm on L(Rⁿ, R^m).

(b) Prove that kT (x)k_R^m ≤ kT k_opkxk_Rⁿ for all x ∈ Rⁿ. (c) Let S ∈ L(R^m, R^p). Prove that kS ◦ T k_op≤ kSk_opkT k_op. (2) For each A ∈ Mmn(R), we define the matrix norm of A to be

kAk = kL_Ak_op

where L_A: Rⁿ→ R^m is the linear map L_A(x) = Ax for any x ∈ Rⁿ. In this exercise, we assume m = n.

(a) Use mathematical induction to prove that kA^kk ≤ kAk^k for any k ≥ 1.

Remark. This implies that the sequence of numbers (kA^kk^1/k) is bounded above by kAk. We define the spectral radius of a square matrix A to be

ρ(A) = lim sup

k→∞

kA^kk^1/k. This implies that ρ(A) ≤ kAk.

(b) Let Eij be the n × n matrix whose ij-th entry is 1 and zero otherwise. Find kE_ijk for all 1 ≤ i, j ≤ n.

(c) Let λ₁, · · · , λ_n be n real numbers. Suppose that D = Pn

i=1λ_iE_ii, i.e. D is a diagonal matrix. Prove that

kDk = max{|λ₁|, · · · , |λ_n|}.

We denote D by diag(λ1, · · · , λn). Prove that ρ(D) = kDk.

(3) Let R > 0. Suppose that the power series f (x) =P∞

k=0a_kx^k is convergent for any x ∈ (−R, R). Here an∈ R for any n ≥ 0. Let A be an n × n real matrix such that kAk ∈ (−R + δ, R − δ) where 0 < δ < R.

(a) Prove that P∞

k=0a_kA^k is convergent in (M_n(R), k · k). In this case, we define f (A) =P∞

k=0akA^k.

(b) If D = diag(λ₁, · · · , λ_n), where λ₁, · · · , λ_n are real numbers in (−R + δ, R − δ).

Prove that f (D) = diag(f (λ1), · · · , f (λn)), i.e. f (D) =Pn

i=1f (λi)Eii.

(c) Let A be diagonalizable¹ with A = SDS⁻¹ where D = diag(λ1, · · · , λn). Sup- pose λ_i ∈ (−R + δ, R − δ) for 1 ≤ i ≤ n. Show that f (A) = Sf (D)S⁻¹.

Remark. If A is diagonalizable with A = SDS⁻¹, then exp(A) = S⁻¹exp(D)S, and cos A = S⁻¹cos(D)S, and sin A = S⁻¹sin(D)S.

(d) Let λ ∈ (−R, R) and denote

J3(λ) =





λ 1 0 0 λ 1 0 0 λ



.

1A matrix A ∈ Mn(R) is said to be diagonalizable if there exists an invertible matrix S and a diagonal matrix D in Mn(R) such that S⁻¹AS = D.

1

2

Compute f (J3(λ)) in terms of f and λ and compute exp(tJ3(λ)) for all t, λ ∈ R.

In general, compute f (J_n(λ)) and compute exp(tJ_n(λ)), where²

Jn(λ) =















λ 1 0 · · · 0 0 λ 1 · · · 0 ... ... ... . .. ...

0 0 0 λ 1

0 0 0 0 λ















n×n

(4) Let A, B : (a, b) → Mn(R) be matrix valued differentiable functions³. . (a) Prove that

d

dtTr(A(t)) = Tr(A⁰(t)) for any a < t < b.

(b) Prove that

(A(t)B(t))⁰ = A⁰(t)B(t) + A(t)B⁰(t) for any a < t < b.

(c) Show that A⁰(t) = 0 for t ∈ (a, b) if and only if there exists A ∈ Mn(R) such that A(t) = A for any a < t < b.

(d) Suppose that A(t) ∈ GLn(R) for all a < t < b. Prove that

(1.1) d

dt(A(t))⁻¹ = −A(t)⁻¹A⁰(t)A(t)⁻¹ for any a < t < b.

(e) Let t0 ∈ (a, b). Let C : [a, b] → M_n(R) be a continuous function. Define F : [a, b] → M_n(R) by

F (t) = Z t

a

C(s)ds, t ∈ [a, b].

Prove that F is differentiable with F⁰ = C.

(f) Let A(t) = e^tcos t e^tsin t

−e^tsin t e^tcos t



, for t ∈ R. Find A⁰(t), Z t

0

A(s)ds and verify the equation (1.1) using A(t).

(5) Let A =

 2 1 1 2



∈ M₂(R).

(a) Let χA(λ) = det(λI2− A). Find the roots of χ_A(λ).

(b) Let λ₁ and λ₂ be the roots of χ_A(λ). Find unit vectors u₁ and u₂ such that Aui = λiui for i = 1, 2 and a matrix S whose i-th column vector is ui with det S > 0. More precisely, if ui= (xi, yi), then

S =

 x1 x2

y₁ y₂

 .

(c) Prove that AS = SD where D = diag(λ1, λ2) and that A is diagonalizable.

(d) Use the result obtained in exercise (3) to compute exp A, cos A and sin A.

2Jnis called a Jordan matrix.

3Let (V, k · k) be a normed space and f : (a, b) → V be a function. (Such a function is called a V -valued function).

Let t0∈ (a, b), we say that f is differentiable at t0 if

t→tlim₀ 1 t − t0

(f (t) − f (t0)) exists.

In this case, we denote the limit by f⁰(t0). If f is differentiable at every point of (a, b), we say that f is differentiable.

We can inductively define f^(k)(t) for any k ≥ 1. (We can also define the right derivative and the left derivative of f.)

3

(e) Solve for the matrix differential equation

X⁰(t) = AX(t), X(0) = I₂. (f) Let Y (t) = cos(tA) and Z(t) = sin(tA)

A for t ≥ 0. Verify that Y (t) and Z(t) are both solutions to the matrix differential equation

Φ⁰⁰(t) + A²Φ(t) = 0.

Here we use

sin θ

θ =

∞

X

k=0

(−1)^k θ^2k

(2k + 1)! for any θ.