4.1.1: Exercises 4.1
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)In Exercises \(\PageIndex{1}\) - \(\PageIndex{6}\), a matrix \(A\) and one of its eigenvectors are given. Find the eigenvalue of \(A\) for the given eigenvector.
\(A=\left[\begin{array}{cc}{9}&{8}\\{-6}&{-5}\end{array}\right]\quad\vec{x}=\left[\begin{array}{c}{-4}\\{3}\end{array}\right]\)
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\(\lambda =3\)
\(A=\left[\begin{array}{cc}{19}&{-6}\\{48}&{-15}\end{array}\right]\quad\vec{x}=\left[\begin{array}{c}{1}\\{3}\end{array}\right]\)
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\(\lambda =1\)
\(A=\left[\begin{array}{cc}{1}&{-2}\\{-2}&{4}\end{array}\right]\quad\vec{x}=\left[\begin{array}{c}{2}\\{1}\end{array}\right]\)
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\(\lambda =0\)
\(A=\left[\begin{array}{ccc}{-11}&{-19}&{14}\\{-6}&{-8}&{6}\\{-12}&{-22}&{15}\end{array}\right]\quad\vec{x}=\left[\begin{array}{c}{3}\\{2}\\{4}\end{array}\right]\)
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\(\lambda =-5\)
\(A=\left[\begin{array}{ccc}{-7}&{1}&{3}\\{10}&{2}&{-3}\\{-20}&{-14}&{1}\end{array}\right]\quad\vec{x}=\left[\begin{array}{c}{1}\\{-2}\\{4}\end{array}\right]\)
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\(\lambda =3\)
\(A=\left[\begin{array}{ccc}{-12}&{-10}&{0}\\{15}&{13}&{0}\\{15}&{18}&{-5}\end{array}\right]\quad\vec{x}=\left[\begin{array}{c}{-1}\\{1}\\{1}\end{array}\right]\)
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\(\lambda =-2\)
In Exercises \(\PageIndex{7}\) – \(\PageIndex{11}\), a matrix \(A\) and one of its eigenvalues are given. Find an eigenvector of \(A\) for the given eigenvalue.
\(A=\left[\begin{array}{cc}{16}&{6}\\{-18}&{-5}\end{array}\right]\quad\lambda =4\)
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\(\vec{x}=\left[\begin{array}{c}{-1}\\{2}\end{array}\right]\)
\(A=\left[\begin{array}{cc}{-2}&{6}\\{-9}&{13}\end{array}\right]\quad\lambda =7\)
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\(\vec{x}=\left[\begin{array}{c}{2}\\{3}\end{array}\right]\)
\(A=\left[\begin{array}{ccc}{-16}&{-28}&{-19}\\{42}&{69}&{46}\\{-42}&{-72}&{-49}\end{array}\right]\quad\lambda =5\)
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\(\vec{x}=\left[\begin{array}{c}{3}\\{-7}\\{7}\end{array}\right]\)
\(A=\left[\begin{array}{ccc}{7}&{-5}&{-10}\\{6}&{2}&{-6}\\{2}&{-5}&{-5}\end{array}\right]\quad\lambda =-3\)
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\(\vec{x}=\left[\begin{array}{c}{1}\\{0}\\{1}\end{array}\right]\)
\(A=\left[\begin{array}{ccc}{4}&{5}&{-3}\\{-7}&{-8}&{3}\\{1}&{-5}&{8}\end{array}\right]\quad\lambda =2\)
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\(\vec{x}=\left[\begin{array}{c}{-1}\\{1}\\{1}\end{array}\right]\)
In Exercises \(\PageIndex{12}\) – \(\PageIndex{28}\), find the eigenvalues of the given matrix. For each eigenvalue, give an eigenvector.
\(\left[\begin{array}{cc}{-1}&{-4}\\{-3}&{-2}\end{array}\right]\)
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\(\lambda_{1}=-5\) with \(\vec{x_{1}}=\left[\begin{array}{c}{1}\\{1}\end{array}\right];\)
\(\lambda_{2}=2\) with \(\vec{x_{2}}=\left[\begin{array}{c}{-4}\\{3}\end{array}\right]\)
\(\left[\begin{array}{cc}{-4}&{72}\\{-1}&{13}\end{array}\right]\)
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\(\lambda_{1}=4\) with \(\vec{x_{1}}=\left[\begin{array}{c}{9}\\{1}\end{array}\right];\)
\(\lambda_{2}=5\) with \(\vec{x_{2}}=\left[\begin{array}{c}{8}\\{1}\end{array}\right]\)
\(\left[\begin{array}{cc}{2}&{-12}\\{2}&{-8}\end{array}\right]\)
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\(\lambda_{1}=-4\) with \(\vec{x_{1}}=\left[\begin{array}{c}{2}\\{1}\end{array}\right];\)
\(\lambda_{2}=-2\) with \(\vec{x_{2}}=\left[\begin{array}{c}{3}\\{1}\end{array}\right]\)
\(\left[\begin{array}{cc}{3}&{12}\\{1}&{-1}\end{array}\right]\)
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\(\lambda_{1}=-3\) with \(\vec{x_{1}}=\left[\begin{array}{c}{-2}\\{1}\end{array}\right];\)
\(\lambda_{2}=5\) with \(\vec{x_{2}}=\left[\begin{array}{c}{6}\\{1}\end{array}\right]\)
\(\left[\begin{array}{cc}{5}&{9}\\{-1}&{-5}\end{array}\right]\)
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\(\lambda_{1}=-4\) with \(\vec{x_{1}}=\left[\begin{array}{c}{-1}\\{1}\end{array}\right];\)
\(\lambda_{2}=4\) with \(\vec{x_{2}}=\left[\begin{array}{c}{-9}\\{1}\end{array}\right]\)
\(\left[\begin{array}{cc}{3}&{-1}\\{-1}&{3}\end{array}\right]\)
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\(\lambda_{1}=2\) with \(\vec{x_{1}}=\left[\begin{array}{c}{1}\\{1}\end{array}\right];\)
\(\lambda_{2}=4\) with \(\vec{x_{2}}=\left[\begin{array}{c}{-1}\\{1}\end{array}\right]\)
\(\left[\begin{array}{cc}{0}&{1}\\{25}&{0}\end{array}\right]\)
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\(\lambda_{1}=-5\) with \(\vec{x_{1}}=\left[\begin{array}{c}{-1}\\{5}\end{array}\right];\)
\(\lambda_{2}=5\) with \(\vec{x_{2}}=\left[\begin{array}{c}{1}\\{5}\end{array}\right]\)
\(\left[\begin{array}{cc}{-3}&{1}\\{0}&{-1}\end{array}\right]\)
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\(\lambda_{1}=-1\) with \(\vec{x_{1}}=\left[\begin{array}{c}{1}\\{2}\end{array}\right];\)
\(\lambda_{2}=-3\) with \(\vec{x_{2}}=\left[\begin{array}{c}{1}\\{0}\end{array}\right]\)
\(\left[\begin{array}{ccc}{1}&{-2}&{-3}\\{0}&{3}&{0}\\{0}&{-1}&{-1}\end{array}\right]\)
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\(\lambda_{1}=-1\) with \(\vec{x_{1}}=\left[\begin{array}{c}{3}\\{0}\\{2}\end{array}\right];\)
\(\lambda_{2}=1\) with \(\vec{x_{2}}=\left[\begin{array}{c}{1}\\{0}\\{0}\end{array}\right]\)
\(\lambda_{3}=3\) with \(\vec{x_{3}}=\left[\begin{array}{c}{5}\\{-8}\\{2}\end{array}\right]\)
\(\left[\begin{array}{ccc}{5}&{-2}&{3}\\{0}&{4}&{0}\\{0}&{-1}&{3}\end{array}\right]\)
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\(\lambda_{1}=3\) with \(\vec{x_{1}}=\left[\begin{array}{c}{-3}\\{0}\\{2}\end{array}\right];\)
\(\lambda_{2}=4\) with \(\vec{x_{2}}=\left[\begin{array}{c}{-5}\\{-1}\\{1}\end{array}\right]\)
\(\lambda_{3}=5\) with \(\vec{x_{3}}=\left[\begin{array}{c}{1}\\{0}\\{0}\end{array}\right]\)
\(\left[\begin{array}{ccc}{1}&{0}&{12}\\{2}&{-5}&{0}\\{1}&{0}&{2}\end{array}\right]\)
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\(\lambda_{1}=-5\) with \(\vec{x_{1}}=\left[\begin{array}{c}{0}\\{1}\\{0}\end{array}\right];\)
\(\lambda_{2}=-2\) with \(\vec{x_{2}}=\left[\begin{array}{c}{-12}\\{-8}\\{3}\end{array}\right]\)
\(\lambda_{3}=5\) with \(\vec{x_{3}}=\left[\begin{array}{c}{15}\\{3}\\{5}\end{array}\right]\)
\(\left[\begin{array}{ccc}{1}&{0}&{-18}\\{-4}&{3}&{-1}\\{1}&{0}&{-8}\end{array}\right]\)
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\(\lambda_{1}=-5\) with \(\vec{x_{1}}=\left[\begin{array}{c}{24}\\{13}\\{8}\end{array}\right];\)
\(\lambda_{2}=-2\) with \(\vec{x_{2}}=\left[\begin{array}{c}{6}\\{5}\\{1}\end{array}\right]\)
\(\lambda_{3}=3\) with \(\vec{x_{3}}=\left[\begin{array}{c}{0}\\{1}\\{0}\end{array}\right]\)
\(\left[\begin{array}{ccc}{-1}&{18}&{0}\\{1}&{2}&{0}\\{5}&{-3}&{-1}\end{array}\right]\)
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\(\lambda_{1}=-4\) with \(\vec{x_{1}}=\left[\begin{array}{c}{-6}\\{1}\\{11}\end{array}\right];\)
\(\lambda_{2}=-1\) with \(\vec{x_{2}}=\left[\begin{array}{c}{0}\\{0}\\{1}\end{array}\right]\)
\(\lambda_{3}=5\) with \(\vec{x_{3}}=\left[\begin{array}{c}{3}\\{1}\\{2}\end{array}\right]\)
\(\left[\begin{array}{ccc}{5}&{0}&{0}\\{1}&{1}&{0}\\{-1}&{5}&{-2}\end{array}\right]\)
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\(\lambda_{1}=-2\) with \(\vec{x_{1}}=\left[\begin{array}{c}{0}\\{0}\\{1}\end{array}\right];\)
\(\lambda_{2}=1\) with \(\vec{x_{2}}=\left[\begin{array}{c}{0}\\{3}\\{5}\end{array}\right]\)
\(\lambda_{3}=5\) with \(\vec{x_{3}}=\left[\begin{array}{c}{28}\\{7}\\{1}\end{array}\right]\)
\(\left[\begin{array}{ccc}{2}&{-1}&{1}\\{0}&{3}&{6}\\{0}&{0}&{7}\end{array}\right]\)
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\(\lambda_{1}=2\) with \(\vec{x_{1}}=\left[\begin{array}{c}{1}\\{0}\\{0}\end{array}\right];\)
\(\lambda_{2}=3\) with \(\vec{x_{2}}=\left[\begin{array}{c}{-1}\\{1}\\{0}\end{array}\right]\)
\(\lambda_{3}=7\) with \(\vec{x_{3}}=\left[\begin{array}{c}{-1}\\{15}\\{10}\end{array}\right]\)
\(\left[\begin{array}{ccc}{3}&{5}&{-5}\\{-2}&{3}&{2}\\{-2}&{5}&{0}\end{array}\right]\)
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\(\lambda_{1}=-2\) with \(\vec{x_{1}}=\left[\begin{array}{c}{1}\\{0}\\{1}\end{array}\right];\)
\(\lambda_{2}=3\) with \(\vec{x_{2}}=\left[\begin{array}{c}{1}\\{1}\\{1}\end{array}\right];\)
\(\lambda_{3}=5\) with \(\vec{x_{3}}=\left[\begin{array}{c}{0}\\{1}\\{1}\end{array}\right]\)