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<h1 class="heading"><a href="linear-algebra-for-team-based-inquiry-learning.html"><span class="title">Linear Algebra for Team-Based Inquiry Learning</span></a></h1>
<p class="byline">Steven Clontz, Drew Lewis</p>
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<main class="main"><div id="content" class="pretext-content"><section xmlns:svg="http://www.w3.org/2000/svg" class="section" id="G3"><h2 class="heading hide-type">
<span class="type">Section</span> <span class="codenumber">5.3</span> <span class="title">Eigenvalues and Characteristic Polynomials (G3)</span>
</h2>
<article class="activity project-like" id="activity-138"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">5.3.1</span><span class="period">.</span>
</h6>
<p id="p-912">An invertible matrix \(M\) and its inverse \(M^{-1}\) are given below:</p>
<div class="displaymath">
\begin{equation*}
M=\left[\begin{array}{cc}1&2\\3&4\end{array}\right]
\hspace{2em}
M^{-1}=\left[\begin{array}{cc}-2&1\\3/2&-1/2\end{array}\right]
\end{equation*}
</div>
<p id="p-913">Which of the following is equal to \(\det(M)\det(M^{-1})\text{?}\)</p>
<ol class="lower-alpha">
<li id="li-653"><p id="p-914">\(\displaystyle -1\)</p></li>
<li id="li-654"><p id="p-915">\(\displaystyle 0\)</p></li>
<li id="li-655"><p id="p-916">\(\displaystyle 1\)</p></li>
<li id="li-656"><p id="p-917">\(\displaystyle 4\)</p></li>
</ol></article><article class="fact theorem-like" id="fact-28"><h6 class="heading">
<span class="type">Fact</span><span class="space"> </span><span class="codenumber">5.3.1</span><span class="period">.</span>
</h6>
<p id="p-918">For every invertible matrix \(M\text{,}\)</p>
<div class="displaymath">
\begin{equation*}
\det(M)\det(M^{-1})= \det(I)=1
\end{equation*}
</div>
<p data-braille="continuation">so \(\det(M^{-1})=\frac{1}{\det(M)}\text{.}\)</p>
<p id="p-919">Furthermore, a square matrix \(M\) is invertible if and only if \(\det(M)\not=0\text{.}\)</p></article><article class="observation remark-like" id="observation-34"><h6 class="heading">
<span class="type">Observation</span><span class="space"> </span><span class="codenumber">5.3.2</span><span class="period">.</span>
</h6>
<p id="p-920">Consider the linear transformation \(A : \IR^2 \rightarrow \IR^2\) given by the matrix \(A = \left[\begin{array}{cc} 2 & 2 \\ 0 & 3 \end{array}\right]\text{.}\)</p>
<pre class="code-block tex2jax_ignore">
\begin{tikzpicture}[scale=0.75]
\fill[red!50] (0,0) rectangle (1,1);
\draw[thin,gray,<->] (-1,0)-- (4,0);
\draw[thin,gray,<->] (0,-1)-- (0,4);
\draw[thick,blue,->] (0,0) -- node[below right] {\(A \vec{e}_1\)}++ (2,0);
\draw[thick,red,->] (0,0) -- node[below] {\(\vec{e}_1\)}++ (1,0);
\draw[thick,blue,->] (0,0) -- node[above left] {\(A \vec{e}_2\)}++(2,3);
\draw[thick,red,->] (0,0) -- node[left] {\(\vec{e}_2\)}++ (0,1);
\draw[blue,dashed] (2,0) -- (4,3) -- (2,3);
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</pre>
<p id="p-921">It is easy to see geometrically that</p>
<div class="displaymath">
\begin{equation*}
A\left[\begin{array}{c}1 \\ 0 \end{array}\right] =
\left[\begin{array}{cc} 2 & 2 \\ 0 & 3 \end{array}\right]\left[\begin{array}{c}1 \\ 0 \end{array}\right]=
\left[\begin{array}{c}2 \\ 0 \end{array}\right]=
2 \left[\begin{array}{c}1 \\ 0 \end{array}\right]\text{.}
\end{equation*}
</div>
<p id="p-922">It is less obvious (but easily checked once you find it) that</p>
<div class="displaymath">
\begin{equation*}
A\left[\begin{array}{c} 2 \\ 1 \end{array}\right] =
\left[\begin{array}{cc} 2 & 2 \\ 0 & 3 \end{array}\right]\left[\begin{array}{c}2 \\ 1 \end{array}\right]=
\left[\begin{array}{c} 6 \\ 3 \end{array}\right] =
3\left[\begin{array}{c} 2 \\ 1 \end{array}\right]\text{.}
\end{equation*}
</div></article><article class="definition definition-like" id="definition-28"><h6 class="heading">
<span class="type">Definition</span><span class="space"> </span><span class="codenumber">5.3.3</span><span class="period">.</span>
</h6>
<p id="p-923">Let \(A \in M_{n,n}\text{.}\) An <dfn class="terminology">eigenvector</dfn> for \(A\) is a vector \(\vec{x} \in \IR^n\) such that \(A\vec{x}\) is parallel to \(\vec{x}\text{.}\)</p>
<pre class="code-block tex2jax_ignore">
\begin{tikzpicture}[scale=0.75]
\fill[gray!50] (0,0) rectangle (1,1);
\draw[thin,gray,<->] (-1,0)-- (4,0);
\draw[thin,gray,<->] (0,-1)-- (0,4);
\draw[thick,blue,->] (0,0) -- node[below right] {\(A \vec{e}_1=2\vec e_1\)}++ (2,0);
\draw[thick,red,->] (0,0) -- node[below] {\(\vec{e}_1\)}++ (1,0);
\draw[thick,gray,->] (0,0) -- node[above left] {\(A \vec{e}_2\)}++(2,3);
\draw[thick,gray,->] (0,0) -- node[left] {\(\vec{e}_2\)}++ (0,1);
\draw[gray,dashed] (2,0) -- (4,3) -- (2,3);
\draw[purple!50!blue,thick,->] (0,0) -- (6,3)
node [below right] {\(
A\left[\begin{array}{c}2\\1\end{array}\right]
=
3\left[\begin{array}{c}2\\1\end{array}\right]
\)};
\draw[purple!50!red,thick,->] (0,0) -- (2,1)
node [above] {\(\left[\begin{array}{c}2\\1\end{array}\right]\)};
\end{tikzpicture}
</pre>
<p id="p-924">In other words, \(A\vec{x}=\lambda \vec{x}\) for some scalar \(\lambda\text{.}\) If \(\vec x\not=\vec 0\text{,}\) then we say \(\vec x\) is a <dfn class="terminology">nontrivial eigenvector</dfn> and we call this \(\lambda\) an <dfn class="terminology">eigenvalue</dfn> of \(A\text{.}\)</p></article><article class="activity project-like" id="activity-139"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">5.3.2</span><span class="period">.</span>
</h6>
<p id="p-925">Finding the eigenvalues \(\lambda\) that satisfy</p>
<div class="displaymath">
\begin{equation*}
A\vec x=\lambda\vec x=\lambda(I\vec x)=(\lambda I)\vec x
\end{equation*}
</div>
<p data-braille="continuation">for some nontrivial eigenvector \(\vec x\) is equivalent to finding nonzero solutions for the matrix equation</p>
<div class="displaymath">
\begin{equation*}
(A-\lambda I)\vec x =\vec 0\text{.}
\end{equation*}
</div>
<p id="p-926">Which of the following must be true for any eigenvalue?</p>
<ol class="lower-alpha">
<li id="li-657"><p id="p-927">The <em class="emphasis">kernel</em> of the transformation with standard matrix \(A-\lambda I\) must contain <em class="emphasis">the zero vector</em>, so \(A-\lambda I\) is <em class="emphasis">invertible</em>.</p></li>
<li id="li-658"><p id="p-928">The <em class="emphasis">kernel</em> of the transformation with standard matrix \(A-\lambda I\) must contain <em class="emphasis">a non-zero vector</em>, so \(A-\lambda I\) is <em class="emphasis">not invertible</em>.</p></li>
<li id="li-659"><p id="p-929">The <em class="emphasis">image</em> of the transformation with standard matrix \(A-\lambda I\) must contain <em class="emphasis">the zero vector</em>, so \(A-\lambda I\) is <em class="emphasis">invertible</em>.</p></li>
<li id="li-660"><p id="p-930">The <em class="emphasis">image</em> of the transformation with standard matrix \(A-\lambda I\) must contain <em class="emphasis">a non-zero vector</em>, so \(A-\lambda I\) is <em class="emphasis">not invertible</em>.</p></li>
</ol></article><article class="fact theorem-like" id="fact-29"><h6 class="heading">
<span class="type">Fact</span><span class="space"> </span><span class="codenumber">5.3.4</span><span class="period">.</span>
</h6>
<p id="p-931">The eigenvalues \(\lambda\) for a matrix \(A\) are the values that make \(A-\lambda I\) non-invertible.</p>
<p id="p-932">Thus the eigenvalues \(\lambda\) for a matrix \(A\) are the solutions to the equation</p>
<div class="displaymath">
\begin{equation*}
\det(A-\lambda I)=0.
\end{equation*}
</div></article><article class="definition definition-like" id="definition-29"><h6 class="heading">
<span class="type">Definition</span><span class="space"> </span><span class="codenumber">5.3.5</span><span class="period">.</span>
</h6>
<p id="p-933">The expression \(\det(A-\lambda I)\) is called <dfn class="terminology">characteristic polynomial</dfn> of \(A\text{.}\)</p>
<p id="p-934">For example, when \(A=\left[\begin{array}{cc}1 & 2 \\ 3 & 4\end{array}\right]\text{,}\) we have</p>
<div class="displaymath">
\begin{equation*}
A-\lambda I=
\left[\begin{array}{cc}1 & 2 \\ 3 & 4\end{array}\right]-
\left[\begin{array}{cc}\lambda & 0 \\ 0 & \lambda\end{array}\right]=
\left[\begin{array}{cc}1-\lambda & 2 \\ 3 & 4-\lambda\end{array}\right]\text{.}
\end{equation*}
</div>
<p id="p-935">Thus the characteristic polynomial of \(A\) is</p>
<div class="displaymath">
\begin{equation*}
\det\left[\begin{array}{cc}1-\lambda & 2 \\ 3 & 4-\lambda\end{array}\right]
=
(1-\lambda)(4-\lambda)-(2)(3)
=
\lambda^2-5\lambda-2
\end{equation*}
</div>
<p data-braille="continuation">and its eigenvalues are the solutions to \(\lambda^2-5\lambda-2=0\text{.}\)</p></article><article class="activity project-like" id="activity-140"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">5.3.3</span><span class="period">.</span>
</h6>
<div class="introduction" id="introduction-41"><p id="p-936">Let \(A = \left[\begin{array}{cc} 5 & 2 \\ -3 & -2 \end{array}\right]\text{.}\)</p></div>
<article class="task exercise-like" id="task-98"><h6 class="heading"><span class="codenumber">(a)</span></h6>
<p id="p-937">Compute \(\det (A-\lambda I)\) to determine the characteristic polynomial of \(A\text{.}\)</p></article><article class="task exercise-like" id="task-99"><h6 class="heading"><span class="codenumber">(b)</span></h6>
<p id="p-938">Set this characteristic polynomial equal to zero and factor to determine the eigenvalues of \(A\text{.}\)</p></article></article><article class="activity project-like" id="activity-141"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">5.3.4</span><span class="period">.</span>
</h6>
<p id="p-939">Find all the eigenvalues for the matrix \(A=\left[\begin{array}{cc} 3 & -3 \\ 2 & -4 \end{array}\right]\text{.}\)</p></article><article class="activity project-like" id="activity-142"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">5.3.5</span><span class="period">.</span>
</h6>
<p id="p-940">Find all the eigenvalues for the matrix \(A=\left[\begin{array}{cc} 1 & -4 \\ 0 & 5 \end{array}\right]\text{.}\)</p></article><article class="activity project-like" id="activity-143"><h6 class="heading">
<span class="type">Activity</span><span class="space"> </span><span class="codenumber">5.3.6</span><span class="period">.</span>
</h6>
<p id="p-941">Find all the eigenvalues for the matrix \(A=\left[\begin{array}{ccc} 3 & -3 & 1 \\ 0 & -4 & 2 \\ 0 & 0 & 7 \end{array}\right]\text{.}\)</p></article><section class="exercises" id="exercises-23"><h3 class="heading hide-type">
<span class="type">Exercises</span> <span class="codenumber">5.3.1</span> <span class="title">Exercises</span>
</h3>
<article class="exercise exercise-like" id="exercise-111"><h6 class="heading"><span class="codenumber">1<span class="period">.</span></span></h6>
<p id="p-942">Explain how to find the eigenvalues of the matrix \(\left[\begin{array}{cc}
-2 & 1 \\
18 & 1
\end{array}\right] \text{.}\)</p>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-111" id="answer-111"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-111"><div class="answer solution-like">
<p id="p-943">The characteristic polynomial of \(\left[\begin{array}{cc}
-2 & 1 \\
18 & 1
\end{array}\right] \) is \(\lambda^{2} + \lambda - 20 \text{.}\)</p>
<p id="p-944">The eigenvalues of \(\left[\begin{array}{cc}
-2 & 1 \\
18 & 1
\end{array}\right] \) are \(-5 \) and \(4 \text{.}\)</p>
</div></div>
</div></article><article class="exercise exercise-like" id="exercise-112"><h6 class="heading"><span class="codenumber">2<span class="period">.</span></span></h6>
<p id="p-945">Explain how to find the eigenvalues of the matrix \(\left[\begin{array}{cc}
3 & 2 \\
2 & 3
\end{array}\right] \text{.}\)</p>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-112" id="answer-112"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-112"><div class="answer solution-like">
<p id="p-946">The characteristic polynomial of \(\left[\begin{array}{cc}
3 & 2 \\
2 & 3
\end{array}\right] \) is \(\lambda^{2} - 6 \lambda + 5 \text{.}\)</p>
<p id="p-947">The eigenvalues of \(\left[\begin{array}{cc}
3 & 2 \\
2 & 3
\end{array}\right] \) are \(1 \) and \(5 \text{.}\)</p>
</div></div>
</div></article><article class="exercise exercise-like" id="exercise-113"><h6 class="heading"><span class="codenumber">3<span class="period">.</span></span></h6>
<p id="p-948">Explain how to find the eigenvalues of the matrix \(\left[\begin{array}{cc}
5 & 2 \\
-10 & -7
\end{array}\right] \text{.}\)</p>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-113" id="answer-113"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-113"><div class="answer solution-like">
<p id="p-949">The characteristic polynomial of \(\left[\begin{array}{cc}
5 & 2 \\
-10 & -7
\end{array}\right] \) is \(\lambda^{2} + 2 \lambda - 15 \text{.}\)</p>
<p id="p-950">The eigenvalues of \(\left[\begin{array}{cc}
5 & 2 \\
-10 & -7
\end{array}\right] \) are \(3 \) and \(-5 \text{.}\)</p>
</div></div>
</div></article><article class="exercise exercise-like" id="exercise-114"><h6 class="heading"><span class="codenumber">4<span class="period">.</span></span></h6>
<p id="p-951">Explain how to find the eigenvalues of the matrix \(\left[\begin{array}{cc}
8 & 2 \\
-15 & -3
\end{array}\right] \text{.}\)</p>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-114" id="answer-114"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-114"><div class="answer solution-like">
<p id="p-952">The characteristic polynomial of \(\left[\begin{array}{cc}
8 & 2 \\
-15 & -3
\end{array}\right] \) is \(\lambda^{2} - 5 \lambda + 6 \text{.}\)</p>
<p id="p-953">The eigenvalues of \(\left[\begin{array}{cc}
8 & 2 \\
-15 & -3
\end{array}\right] \) are \(2 \) and \(3 \text{.}\)</p>
</div></div>
</div></article><article class="exercise exercise-like" id="exercise-115"><h6 class="heading"><span class="codenumber">5<span class="period">.</span></span></h6>
<p id="p-954">Explain how to find the eigenvalues of the matrix \(\left[\begin{array}{cc}
7 & 1 \\
-18 & -4
\end{array}\right] \text{.}\)</p>
<div class="solutions">
<a data-knowl="" class="id-ref answer-knowl original" data-refid="hk-answer-115" id="answer-115"><span class="type">Answer</span></a><div class="hidden-content tex2jax_ignore" id="hk-answer-115"><div class="answer solution-like">
<p id="p-955">The characteristic polynomial of \(\left[\begin{array}{cc}
7 & 1 \\
-18 & -4
\end{array}\right] \) is \(\lambda^{2} - 3 \lambda - 10 \text{.}\)</p>
<p id="p-956">The eigenvalues of \(\left[\begin{array}{cc}
7 & 1 \\
-18 & -4
\end{array}\right] \) are \(5 \) and \(-2 \text{.}\)</p>
</div></div>
</div></article><p id="p-957"><em class="emphasis">Additional exercises available at <a class="external" href="https://checkit.clontz.org/banks/tbil-la/G3/" target="_blank">checkit.clontz.org</a>.</em></p></section></section></div></main>
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