MATH 308 Lecture 29

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Eigenvalues and Eigenvectors

(See MATH 323 Lecture 24#Eigenvalues and Eigenvectors→)

Just to mention it here,

Eigenvalues $\lambda$ are solutions to $|A-\lambda I|=0$
Eigenvectors are nonzero solutions $\overrightarrow{x}$ to $(A-\lambda I)\overrightarrow{x}=0\to AX=\lambda X$

Exercise 5

Find eigenvalues and eigenvectors of $A=\left[\begin{array}{cc}5& -1\\ 3& 1\end{array}\right]$

$\begin{array}{rl}\left|\begin{array}{cc}5-\lambda & -1\\ 3& 1-\lambda \end{array}\right|=5-6\lambda +{\lambda }^2+3& =0\\ \lambda & =2,4\\ \left[\begin{array}{cc}3& -1\\ 3& -1\end{array}\right]{\overrightarrow{x}}_1& =\overrightarrow{0}\\ \left[\begin{array}{cc}-3& 1\\ 0& 0\end{array}\right]{\overrightarrow{x}}_1& =\overrightarrow{0}\\ {\overrightarrow{x}}_1& =\alpha ⟨1,3⟩\\ \left[\begin{array}{cc}1& -1\\ 3& -3\end{array}\right]{\overrightarrow{x}}_2& =\overrightarrow{0}\\ \left[\begin{array}{cc}1& -1\\ 0& 0\end{array}\right]{\overrightarrow{x}}_2& =\overrightarrow{0}\\ {\overrightarrow{x}}_2& =\alpha \end{array}$

Find eigenvalues and eigenvectors of $B=\left[\begin{array}{ccc}1& 0& 0\\ 2& 1& 2\\ 3& 2& 1\end{array}\right]$

$\begin{array}{rl}\left|\begin{array}{ccc}1-\lambda & 0& 0\\ 2& 1-\lambda & 2\\ 3& 2& 1-\lambda \end{array}\right|\to \lambda & =-1,1,3\\ {\overrightarrow{x}}_2& =⟨0,1,-1⟩\\ {\overrightarrow{x}}_1& =⟨2,-3,-2⟩\\ {\overrightarrow{x}}_3& =⟨0,1,1⟩\\ \end{array}$

Solving Systems of Equations

Theorem 7.4.1

Let $X_1,\dots ,X_p$ be $p$ solutions to the homogeneous system

$X^{\prime }(t)=A(t)X(t)$

on the interval $I$, where $A(t)$ is a continuous $n\times n$ matrix function on $I$,

then for any real numbers $c_1,\dots ,cp$, the vector

$\overrightarrow{x}(t)=\sum _{i=1}^p{c}_i{X}_i$

is a solution to the homogeneous system.

Theorem 7.4.2

Let $X_1,\dots ,X_n$ be $n$ linearly independent solutions to the homogeneous system

$X^{\prime }(t)=A(t)X(t)$

on the interval $I$, where $A(t)$ is a continuous $n\times n$ matrix function on $I$,

Any solution $\overrightarrow{x}(t)$ can be expressed in the form

$\overrightarrow{x}(t)=\sum _{i=1}^p{c}_i{X}_i$

where $c_1,\dots ,c_n$ are constants.

Theorem 7.4.3

The Wronskian $W(X_1,\dots ,X_n)$ is either 0 on $I$ or never vanishes on $I$.

In other words, if $W(X_1,\dots ,X_n)=0$, then the solutions are linearly dependent and do not describe all possible solutions.

Otherwise, we have found all solutions