LEC DQN 1

Recall that the general solution for the first-order linear differential equation $\dot{x}=ax$ is $x(t)=c{e}^{at}$.

Definition 384.1(Definition).

1. An ordinary DE is an equation containing an unknown function of one variable real/complex variable $x$ and its derivatives.
2. A Linear DE is a DE that is linear in the unknown function and its derivatives.
3. The order of a DE is the highest order of derivative of the unknown function that appears in the DE.
4. A system of DEs is said to be uncoupled if each DE depends on only one variable.
5. A system of DEs is said to be autonomous if there is no explicit dependence on the independent variable (for example, time).
6. Initial conditions which result in a constant solution are called equilibrium points.

Exponentials of operators

A system of linear first order differential equations can be expressed as $\dot{\mathbf{x}}=A\mathbf{x}$. Solving this system is equivalent to finding the “exponential matrix” $e^{At}$, which we will do now.

Recall the properties of the operator norm.

Theorem 384.2(Theorem).

Given $T\in \mathcal{L}({\mathbb{R}}^n)$ and $t_0>0$, the series

$$\sum _{k=0}^{\infty }\frac{T^k{t}^k}{k!}$$

is absolutely and uniformly convergent for all $|t|⩽t_0$.

The exponential of the linear operator $T$ is defined by the absolutely convergent series

$$e^T:=\sum _{k=0}^{\infty }\frac{T^k}{k!}.$$

Definition 384.3(Definition).

Let $A$ be an $n\times n$ matrix. Then for $t\in \mathbb{R}$,

$$e^{At}:=\sum _{k=0}^{\infty }\frac{A^k{t}^k}{k!}.$$

Proposition 384.4(Proposition).

If $P$ and $T$ are linear transformations on ${\mathbb{R}}^n$ and $S=PT{P}^{-1}$, then $e^S=P{e}^T{P}^{-1}$.

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Corollary 384.5(Corollary).

If $A=P\text{diag}[{\lambda }_j]P^{-1}$, then $e^{At}=P\text{diag}[e^{{\lambda }_{j}t}]P^{-1}$.

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Proposition 384.6(Proposition).

If $S$ and $T$ are linear transformations on ${\mathbb{R}}^n$ which commute, then $e^{S+T}=e^S{e}^T$.

Corollary 384.7(Corollary).

If $T$ is a linear transformation on ${\mathbb{R}}^n$, the inverse of the linear transformation $e^T$ is given by $(e^T{)}^{-1}=e^{-T}$.

Corollary 384.8(Corollary).

1. If $A=\left[\begin{array}{cc}a& -b\\ b& a\end{array}\right]$, then $e^A=e^a\left[\begin{array}{cc}\cos b& -\sin b\\ \sin b& \cos b\end{array}\right]$.
2. If $A=\left[\begin{array}{cc}a& b\\ 0& a\end{array}\right]$, then $e^A=e^a\left[\begin{array}{cc}1& b\\ 0& 1\end{array}\right]$.

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Proposition 384.9(Proposition).

Let $A$ be any $2\times 2$ matrix. There exists an invertible $2\times 2$ matrix $P$ (whose columns consist of generalized eigenvectors of $A$) such that the matrix $B=P^{-1}AP$ has one of the following forms:

$$B=\left[\begin{array}{cc}\lambda & \\ & \mu \end{array}\right],B=\left[\begin{array}{cc}\lambda & 1\\ & \lambda \end{array}\right],\text{or}B=\left[\begin{array}{cc}a& -b\\ b& a\end{array}\right].$$

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We can now compute the matrix $e^{At}$ for any $2\times 2$ matrix $A$. Let $B$ be given by Prp 9. Then, by Cor 5 and Cor 8,

$$e^{Bt}=\left[\begin{array}{cc}{e}^{\lambda t}& \\ & e^{\mu t}\end{array}\right],e^{Bt}=e^{\lambda t}\left[\begin{array}{cc}1& t\\ & 1\end{array}\right],\text{or}{e}^{Bt}=e^{at}\left[\begin{array}{cc}\cos bt& -\sin bt\\ \sin bt& \cos bt\end{array}\right].$$

By Prp 4, $e^{At}$ is given by

$$e^{At}=P{e}^{Bt}{P}^{-1}.$$

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The fundamental theorem for linear systems

Lemma 384.10(Lemma).

Let $A$ be a square matrix, then

$$\frac{d}{dt}{e}^{At}=A{e}^{At}.$$

Theorem 384.11(The fundamental theorem for linear systems).

Let $A$ be an $n\times n$ matrix. Then for a given ${\mathbf{x}}_0\in {\mathbb{R}}^n$, the initial value problem

$$\begin{array}{r}\dot{\mathbf{x}}=A\mathbf{x},\mathbf{x}(0)={\mathbf{x}}_0\end{array}$$

has a unique solution by

$$\mathbf{x}(t)=e^{At}{\mathbf{x}}_0.$$

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Computing exponential matrices

Complex eigenvalues:

Theorem 384.12(Theorem).

If the $2n\times 2n$ real matrix $A$ has $2n$ distinct complex eigenvalues ${\lambda }_j=a_j+i{b}_j$ and ${\overline{\lambda }}_j=a_j-i{b}_j$ and corresponding eigenvectors ${\mathbf{w}}_j={\mathbf{u}}_j+i{\mathbf{v}}_j$ and ${\overline{\mathbf{w}}}_j={\mathbf{u}}_j-i{\mathbf{v}}_j$, then $\{{\mathbf{v}}_1,{\mathbf{u}}_1,\dots ,{\mathbf{v}}_n,{\mathbf{u}}_n\}$ is a basis for ${\mathbb{R}}^{2n}$ and the matrix

$$P=\left[\begin{array}{ccccccc}{\mathbf{v}}_1& {\mathbf{u}}_1& {\mathbf{v}}_2& {\mathbf{u}}_2& \dots & {\mathbf{v}}_n& {\mathbf{u}}_n\end{array}\right]$$

is invertible and

$$P^{-1}AP=\text{diag}\left[\begin{array}{cc}{a}_j& -b_j\\ b_j& a_j\end{array}\right].$$

Multiple real eigenvectors

Theorem 384.13(Theorem).

Let $A$ be a real $n\times n$ matrix with real eigenvalues ${\lambda }_1,\dots ,{\lambda }_n$ repeated according to their multiplicity. Then there exists a basis of generalized eigenvectors for ${\mathbb{R}}^n$. If $\{{\mathbf{v}}_1,\dots ,{\mathbf{v}}_n\}$ is any basis of generalized eigenvectors for ${\mathbb{R}}^n$, the matrix $P=\left[\begin{array}{ccc}{\mathbf{v}}_1& \dots & {\mathbf{v}}_n\end{array}\right]$ is invertible,

$$A=S+N$$

where $P^{-1}SP=\text{diag}[{\lambda }_j]$, and $N=A-S$ is nilpotent of order $k⩽n$, and $S$ and $N$ commute.

Multiple complex eigenvalues

Theorem 384.14(Theorem).

Let $A$ be a real $2n\times 2n$ matrix with complex eigenvalues ${\lambda }_j=a_j+i{b}_j$ and ${\overline{\lambda }}_j=a_j-i{b}_j$ for $j=1,\dots ,n$. Then there exist generalized eigenvectors ${\mathbf{w}}_j={\mathbf{u}}_j+i{\mathbf{v}}_j$ and ${\overline{\mathbf{w}}}_j={\mathbf{u}}_j-i{\mathbf{v}}_j$ for $i=1,\dots ,n$ such that $\{{\mathbf{v}}_1,{\mathbf{u}}_1,\dots ,{\mathbf{v}}_n,{\mathbf{u}}_n\}$ is a basis for ${\mathbb{R}}^{2n}$. For any such basis, the matrix $P=\left[\begin{array}{ccccc}{\mathbf{v}}_1& {\mathbf{u}}_1& \dots & {\mathbf{v}}_n& {\mathbf{u}}_n\end{array}\right]$ is invertible, and

$$A=S+N$$

where

$$P^{-1}SP=\text{diag}\left[\begin{array}{cc}{a}_j& -b_j\\ b_j& a_j\end{array}\right],$$

the matrix $N=A-S$ is nilpotent of order $k⩽2n$, and $S$ and $N$ commute.