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Linear Algebra in Action

Harry Dym

Chapter 13

Difference equations and differential equations - all with Video Answers

Educators

Chapter Questions

Problem 1

Show that if $J=\operatorname{diag}\left\{\lambda_1, \ldots, \lambda_p\right\}, V=\left[\mathbf{v}_1 \cdots \mathbf{v}_p\right]$ and $\left(V^{-1}\right)^T=\left[\begin{array}{lll}\mathbf{w}_1 & \cdots & \mathbf{w}_p\end{array}\right]$, then the solution (13.2) of the system (13.1) can be expressed in the form
$$
\mathbf{x}_n=\sum_{j=1}^p \lambda_j^n \mathbf{v}_j \mathbf{w}_j^T \mathbf{x}_0 .
$$

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Problem 2

Show that if, in the setting of Exercise 13.1, $\left|\lambda_1\right|>\left|\lambda_j\right|$ for $j=2, \ldots, p$, then
$$
\lim _{n \uparrow \infty} \frac{1}{\lambda_1^n} \mathbf{x}_n=\mathbf{v}_1 \mathbf{w}_1^T \mathbf{x}_0 .
$$

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Problem 3

The output $\mathbf{u}_n$ of a chemical plant at time $n, n=0,1, \ldots$, is modelled by a system of the form $\mathbf{u}_n=A^n \mathbf{u}_0$. Show that if
$$
A=\left[\begin{array}{rrc}
1 & -3 / 2 & 0 \\
0 & 1 / 2 & 0 \\
0 & 0 & 1 / 4
\end{array}\right] \text { and } \mathbf{u}_0=\left[\begin{array}{l}
a \\
b \\
c
\end{array}\right] \text {, then } \lim _{n \rightarrow \infty} \mathbf{u}_n=\left[\begin{array}{c}
a-3 b \\
0 \\
0
\end{array}\right] \text {. }
$$

Victor Salazar
Victor Salazar
Numerade Educator
16:56

Problem 4

Find an explicit formula for the solution $\mathbf{u}_n$ of the system $\mathbf{u}_n=A^n \mathbf{u}_0$ when
$$
A=\left[\begin{array}{lll}
1 & 2 & 0 \\
0 & 1 & 3 \\
0 & 0 & 1
\end{array}\right] \quad \text { and } \mathbf{u}_0=\left[\begin{array}{l}
2 \\
3 \\
0
\end{array}\right] \text {. }
$$

Anthony Ramos
Anthony Ramos
Numerade Educator
03:18

Problem 5

Calculate $V^{-1} \mathbf{x}_0$ when $V=\left[\begin{array}{lll}6 & 2 & 2 \\ 0 & 3 & 1 \\ 0 & 0 & 1\end{array}\right]$ and $\mathbf{x}_0=\left[\begin{array}{l}6 \\ 0 \\ 0\end{array}\right]$ both directly (i.e., by first calculating $V^{-1}$ and then calculating the product $V^{-1} \mathbf{x}_0$ ) and indirectly by solving the equation $V \mathbf{y}_0=\mathrm{x}_0$, and compare the effort.

Naresh Bagrecha
Naresh Bagrecha
Numerade Educator

Problem 6

Verify the bound (13.4).

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Problem 7

Show that if $A \in \mathbb{C}^{p \times p}$, then the partial sums
$$
S_k=\sum_{j=0}^k \frac{A^k}{k!}
$$
form a Cauchy sequence in the normed linear space $\mathbb{C}^{p \times p}$ with respect to any multiplicative norm on that space.

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Problem 8

Show that if $A \in \mathbb{C}^{p \times p}$, then
$$
\left\|\frac{e^{h A}-I_p-h A}{h}\right\| \leq \frac{e^{|h|\|A\|}-1-|h|\|A\|}{|h|} \leq\left(e^{|h|\|A\|}-1\right)\|A\| .
$$

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Problem 9

Show that if $A, B \in \mathbb{C}^{p \times p}$ and $A B=B A$, then
$$
e^{A+B}=e^A e^B \text {. }
$$

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02:18

Problem 10

Exhibit a pair of matrices $A, B \in \mathbb{C}^{p \times p}$ such that $e^{A+B} \neq$ $e^A e^B$.

David Mccaslin
David Mccaslin
Numerade Educator
04:02

Problem 11

Show that if $A, B \in \mathbb{C}^{p \times p}$, then
$$
\lim _{(s, t) \rightarrow(0,0)} \frac{e^{t A} e^{s B} e^{-t A} e^{-s B}-I_p}{s t}=A B-B A \text {. }
$$

Let
$$
F(t)=e^{t A}=I_p+t A+t^2 \frac{A^2}{2!}+\cdots .
$$

Then
$$
F(0)=I_p
$$
and
$$
\begin{aligned}
\frac{F(t+h)-F(t)}{h} & =\frac{e^{(t+h) A}-e^{t A}}{h} \\
& =e^{t A}\left(\frac{e^{h A}-I_p}{h}\right),
\end{aligned}
$$
which tends to
$$
e^{t A} A=A e^{t A}
$$
as $h$ tends to zero, thanks to the bound (13.6). Thus, the derivative
$$
F^{\prime}(t)=\lim _{h \rightarrow 0} \frac{F(t+h)-F(t)}{h}=A F(t) .
$$

The same definition is used for the derivative of any suitably smooth matrix valued function $F(t)=\left[f_{i j}(t)\right]$ with entries $f_{i j}(t)$ and implies that
$$
F^{\prime}(t)=\left[f_{i j}^{\prime}(t)\right], \quad \text { and correspondingly } \quad \int_a^b F(s) d s=\left[\int_a^b f_{i j}(s) d s\right] ;
$$
i.e., differentiation and integration of a matrix valued function is carried out on each entry in the matrix separately.

Brandon Collins
Brandon Collins
Numerade Educator

Problem 12

Show that if $F(t)$ is an invertible suitably smooth $p \times p$ matrix valued function on the interval $a<t<b$, then
$$
\lim _{h \rightarrow 0} \frac{F(t+h)^{-1}-F(t)^{-1}}{h}=-F(t)^{-1} F^{\prime}(t) F(t)^{-1} \text { for } a<t<b .
$$

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Problem 13

Calculate $e^A$ when $A=\left[\begin{array}{ll}0 & b \\ c & 0\end{array}\right]$.

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Problem 14

Calculate $e^A$ when $A=\left[\begin{array}{ll}a & b \\ b & a\end{array}\right]$

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Problem 15

Calculate $e^A$ when $A=\left[\begin{array}{cc}a & b \\ -b & a\end{array}\right]$.

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06:06

Problem 16

Verify formula (13.12).

Amany Waheeb
Amany Waheeb
Numerade Educator
04:23

Problem 17

Show that if $J=\operatorname{diag}\left\{\lambda_1, \ldots, \lambda_p\right\}, V=\left[\mathbf{v}_1 \cdots \mathbf{v}_p\right]$ and $\left(V^{-1}\right)^T=\left[\begin{array}{lll}\mathbf{w}_1 & \cdots & \mathbf{w}_p\end{array}\right]$, then the solution (13.8) of the system (13.9) can be expressed in the form
$$
\mathbf{x}(t)=\sum_{j=1}^p e^{(t-a) \lambda_j} \mathbf{v}_j \mathbf{w}_j^T \mathbf{x}(a) .
$$

Christian Otero
Christian Otero
Numerade Educator

Problem 18

Show that if, in the setting of Exercise 13.17, $\left|\lambda_1\right|>\left|\lambda_j\right|$ for $j=2, \ldots, p$, then
$$
\lim _{t \uparrow \infty} e^{-t \lambda_1} \mathbf{x}(t)=e^{-a \lambda_1} \mathbf{v}_1 \mathbf{w}_1^T \mathbf{x}(a) .
$$

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00:43

Problem 19

Give an explicit formula for $e^{t A}$ when
$$
A=\left[\begin{array}{ccc}
0 & 1 & 0 \\
-1 & 0 & 1 \\
0 & -1 & 0
\end{array}\right]
$$

Victor Salazar
Victor Salazar
Numerade Educator
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Problem 20

Let $A=V J V^{-1}$, where $J=\left[\begin{array}{lll}2 & 1 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3\end{array}\right], V=\left[\begin{array}{lll}\mathbf{v}_1 & \mathbf{v}_2 & \mathbf{v}_3\end{array}\right]$ and $\left(V^T\right)^{-1}=\left[\begin{array}{lll}\mathbf{w}_1 & \mathbf{w}_2 & \mathbf{w}_3\end{array}\right]$. Evaluate the limit of the matrix valued function $e^{-3 t} e^{t A}$ as $t \uparrow \infty$.

Victor Salazar
Victor Salazar
Numerade Educator

Problem 21

Let $h(t)$ be a continuous real-valued function on the interval $a \leq t \leq b$. Show that
$$
\begin{aligned}
& \int_a^t h\left(s_2\right)\left\{\int_a^{s_2} h\left(s_1\right) d s_1\right\} d s_2=\left(\int_a^t h(s) d s\right)^2 / 2!, \\
& \int_a^t h\left(s_3\right)\left[\int_a^{s_3} h\left(s_2\right)\left\{\int_a^{s_2} h\left(s_1\right) d s_1\right\} d s_2\right] d s_3=\left(\int_a^t h(s) d s\right)^3 / 3!\text {, } \\
&
\end{aligned}
$$
etc.

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Problem 22

(Gronwall's inequality) Let $\alpha>0$ and let $u(t)$ and $h(t)$ be continuous real-valued functions on the interval $a \leq t \leq b$ such that
$$
u(t) \leq \alpha+\int_a^t h(s) u(s) d s \text { and } h(t) \geq 0 \text { for } \quad a \leq t \leq b .
$$

Show that
$$
u(t) \leq \alpha \exp \left(\int_a^t h(s) d s\right) \quad \text { for } \quad a \leq t \leq b
$$

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Problem 23

Let $B \in \mathbb{C}^{p \times p}$. Show that if $B$ is skew-Hermitian, then $e^B$ is unitary.

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03:20

Problem 24

Let $F(t)=e^{t B}$, where $B \in \mathbb{R}^{p \times p}$. Show that $F(t)$ is an orthogonal matrix for every $t \in \mathbb{R}$ if and only if $B$ is skew-symmetric.

Chris Trentman
Chris Trentman
Numerade Educator
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Problem 25

Let $B \in \mathbb{R}^{p \times p}$ and let $\mathbf{x}(t), t \geq 0$, denote the solution of the differential equation $\mathbf{x}^{\prime}(t)=B \mathbf{x}(t)$ for $t \geq 0$ that meets the initial condition $\mathbf{x}(0)=\mathbf{c} \in \mathbb{R}^p$.
(a) Show that $\frac{d}{d t}\|\mathbf{x}(t)\|^2=\mathbf{x}(t)^T\left(B+B^T\right) \mathbf{x}(t)$ for every $t \geq 0$.
(b) Show that if $B$ is skew-symmetric, then $\|\mathbf{x}(t)\|=\|\mathbf{x}(0)\|$ for every $t \geq 0$.

Victor Salazar
Victor Salazar
Numerade Educator

Problem 26

Let $A \in \mathbb{R}^{p \times p}$ and $U(t), t \geq 0$, be a one-parameter family of $p \times p$ real matrices such that $U^{\prime}(t)=B(t) U(t)$ for $t>0$ and $U(0)=I_p$. Show that $F(t)=U(t) A U(t)^{-1}$ is a solution of the differential equation
$$
F^{\prime}(t)=B(t) F(t)-F(t) B(t) \quad \text { for } t \geq 0 .
$$

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Problem 27

Show that if $F(t)$ is the only smooth solution of a differential equation of the form (13.13) with suitably smooth $B(t)$, then $F(t)=$ $U(t) F(0) U(t)^{-1}$ for $t \geq 0$. [HINT: Consider $U(t) F(0) U(t)^{-1}$ when $U(t)$ is a solution of $U^{\prime}(t)=B(t) U(t)$ with $U(0)=I_p$.]

A pair of matrix valued functions $F(t)$ and $B(t)$ that are related by equation (13.13) is said to be a Lax pair, and the solution $F(t)=U(t) F(0) U(t)^{-1}$ is said to be isospectral because its eigenvalues are independent of $t$.

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Problem 28

Show by example that item (b) in the list just above is not necessarily correct if the assumption that $J$ is a diagonal matrix is dropped.

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00:51

Problem 29

Show by example that item (d) in the list just above is not necessarily correct if the assumption that $J$ is a diagonal matrix is dropped.

Fuzail Shakir
Fuzail Shakir
Numerade Educator
04:47

Problem 30

Show that if $A$ is a companion matrix, then, in the notation of Theorem 5.11,
$A$ is invertible $\Longleftrightarrow \lambda_1 \cdots \lambda_k \neq 0 \Longleftrightarrow a_0 \neq 0$.

Chris Trentman
Chris Trentman
Numerade Educator

Problem 31

Find an explicit formula for $x_n$, for $n=0,1, \ldots$, given that $x_0=-1, x_1=2$ and $x_{k+1}=3 x_k-2 x_{k-1}$ for $k=1,2, \ldots$.

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Problem 32

The Fibonacci sequence $x_n, n=0,1, \ldots$, is prescribed by the initial conditions $x_0=1, x_1=1$ and the difference equation $x_{n+1}=$ $x_n+x_{n-1}$ for $n=1,2, \ldots$. Find an explicit formula for $x_n$ and use it to calculate the golden mean, $\lim _{n \dagger \infty} x_n / x_{n+1}$.

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01:33

Problem 33

Find the solution of the third-order difference equation
$$
x_{n+3}=3 x_{n+2}-3 x_{n+1}+x_n, n=0,1, \ldots
$$
subject to the initial conditions $x_0=1, x_1=2$ and $x_2=8$.

Adriano Chikande
Adriano Chikande
Numerade Educator
01:13

Problem 34

Find the solution of the third-order differential equation
$$
x^{(3)}(t)=3 x^{(2)}(t)-3 x^{(1)}(t)+x(t), t \geq 0,
$$
subject to the initial conditions
$$
x(0)=1, x^{(1)}(0)=2, x^{(2)}(0)=8 \text {. }
$$

Anand Jangid
Anand Jangid
Numerade Educator

Problem 35

Let $\mathbf{u}^{\prime}(t)=\left[\begin{array}{ll}0 & \alpha \\ \alpha & 0\end{array}\right] \mathbf{u}(t)$ for $t \geq 0$. Show in two different ways that $\|\mathbf{u}(t)\|_2=\|\mathbf{u}(0)\|_2$ if $\alpha+\bar{\alpha}=0$ : first by showing that the derivative of $\|\mathbf{u}(t)\|_2$ with respect to $t$ is constant and then by invoking Exercise 13.23 .

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Problem 36

In the setting of Exercise 13.35, describe $\|\mathbf{u}(t)\|_2$ as $t \dagger \infty$ if $\alpha+\bar{\alpha} \neq 0$.

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18:16

Problem 37

Evaluate $\lim _{t \uparrow \infty} t^{-2} e^{-2 t} \mathbf{y}(t)$ for the solution $\mathbf{y}(t)$ of the equation
$$
\mathbf{y}^{\prime}(t)=\left[\begin{array}{ccc}
0 & 1 & 0 \\
0 & 0 & 1 \\
8 & -12 & 6
\end{array}\right] \mathbf{y}(t), t \geq 0, \text { when } \mathbf{y}(0)=\left[\begin{array}{l}
8 \\
8 \\
8
\end{array}\right] .
$$

Bryan Lynn
Bryan Lynn
Numerade Educator

Problem 38

Let $u_1(t), \ldots, u_p(t)$ be solutions of the homogeneous differential equation considered in Lemma 13.10. Show that the vectors $\mathbf{u}_j(t)$ that are defined by the formulas $\mathbf{u}_j(t)^T=\left[\begin{array}{llll}u_j(t) & u_j^{(1)}(t) & \cdots & u_j^{(p-1)}(t)\end{array}\right]$ for $j=1, \ldots, p$ are linearly independent at one point in the interval $\alpha \leq t \leq$ $\beta$ if and only if they are linearly independent at every point in the interval.

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01:11

Problem 39

Verify that the function $u(t)$ specified in formula (13.28) is a solution of the differential equation (13.26) for every choice of the constants $d_1(\alpha)$ and $d_2(\alpha)$.

Manik Pulyani
Manik Pulyani
Numerade Educator
09:23

Problem 40

Show that if the arbitrary constants in formula (13.28) are specified as $d_2(\alpha)=0$ and $d_1(\alpha)=\gamma^{-1} \int_\alpha^\beta a(s)^{-1} p(s) u_2(s) g(s) d s$, then the solution can be expressed as
$$
u(t)=\int_\alpha^\beta G(t, s) \frac{p(s)}{\gamma a(s)} g(s) d s,
$$
where
$$
G(t, s)=\left\{\begin{array}{lll}
u_1(t) u_2(s) & \text { if } & \alpha \leq t \leq s \leq \beta \\
u_1(s) u_2(t) & \text { if } & \alpha \leq s \leq t \leq \beta
\end{array} .\right.
$$
(The kernel $G(t, s)$ is called the Green function of the problem.)

Mike Gaerlan
Mike Gaerlan
Numerade Educator

Problem 41

Use the formulas in Exercise 13.40 to show that for any choice of $a \in \mathbb{R}$ and $b \in \mathbb{R}$, there exist a pair of constants $\kappa_1$ and $\kappa_2$ such that $a u(\alpha)+b u^{\prime}(\alpha)=\kappa_1\left(a u_1(\alpha)+b u_1^{\prime}(\alpha)\right)$ and $a u(\beta)+b u^{\prime}(\beta)=\kappa_2\left(a u_2(\beta)+\right.$ $\left.b u_2^{\prime}(\beta)\right)$.

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