Notes

 

Eigenvalues and Eigenvector Notes
\((A-\lambda I)\vec{v}=\vec{0}\)

 

 

Practice Problems

\(\textbf{1)}\) Find the eigenvector of \(\left[ {\begin{array}{cc}2 & -12 \\-3 & 2 \end{array} } \right] \text{ where } \lambda=8\)

Show Eigenvector

The eigenvector for \(\lambda=8\) is \(\vec{v}=\left[ {\begin{array}{cc}-2 \\1 \end{array} } \right]\)

 

\(\textbf{2)}\) Find the eigenvector of \(\left[ {\begin{array}{cc}2 & -12 \\-3 & 2 \end{array} } \right] \text{ where } \lambda=-4\)

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The eigenvector for \(\lambda=-4\) is \(\vec{v}=\left[ {\begin{array}{cc}2 \\1 \end{array} } \right]\)

 

\(\textbf{3)}\) Find the eigenvector of \(\left[ {\begin{array}{cc}1 & 2 \\11 & 10 \end{array} } \right] \text{ where } \lambda=12\)

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The eigenvector for \(\lambda=12\) is \(\vec{v}=\left[ {\begin{array}{cc}2 \\11 \end{array} } \right]\)

 

\(\textbf{4)}\) Find the eigenvector of \(\left[ {\begin{array}{cc}1 & 2 \\11 & 10 \end{array} } \right] \text{ where } \lambda=-1\)

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The eigenvector for \(\lambda=-1\) is \(\vec{v}=\left[ {\begin{array}{cc}-1 \\1 \end{array} } \right]\)

 

\(\textbf{5)}\) Find the eigenvector of \(\left[ {\begin{array}{ccc}5 & -5 & -3 \\3 & -3 & -6 \\1 & -1 & 0 \\\end{array} } \right] \text{ where } \lambda=3\)

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The eigenvector for \(\lambda=3\) is \(\vec{v}=\left[ {\begin{array}{cc}4 \\1 \\1 \end{array} } \right]\)

 

\(\textbf{6)}\) Find the eigenvector of \(\left[ {\begin{array}{ccc}5 & -5 & -3 \\3 & -3 & -6 \\1 & -1 & 0 \\\end{array} } \right] \text{ where } \lambda=-1\)

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The eigenvector for \(\lambda=3\) is \(\vec{v}=\left[ {\begin{array}{cc}8 \\9 \\1 \end{array} } \right]\)

 

See Related Pages\(\)

\(\bullet\text{ Intro to Matrices}\)
\(\,\,\,\,\,\,\,\,\)\(\left[ {\begin{array}{ccc}4 & -5 & 2 \\1 & 0 & 3 \\\end{array} } \right]\)

\(\bullet\text{ Matrix Operations}\)
\(\,\,\,\,\,\,\,\,\)\( \left[ {\begin{array}{ccc}3 & 45 & 6 \\-8 & 2 & 4 \\1 & 0 & 3 \\\end{array} } \right]\)\(+\left[ {\begin{array}{ccc}3 & 45 & 6 \\-8 & 2 & 4 \\1 & 0 & 3 \\\end{array} } \right]\)

\(\bullet\text{ Multiplying Matrices}\)
\(\,\,\,\,\,\,\,\,\)\(\left[{\begin{array}{cc} 1 & 2 \\ -3 & -4 \\\end{array} } \right]\)\(\left[ {\begin{array}{cc}6 & -3 \\5 & 0 \\\end{array} } \right]\)

\(\bullet\text{ Determinants}\)
\(\,\,\,\,\,\,\,\,\)\(\left|{\begin{array}{cc} a & b \\ c & d \\ \end{array} } \right|=ad-bc\)

\(\bullet\text{ Cramer’s Rule}\)
\(\,\,\,\,\,\,\,\,\text{ax+by=e } \& \text{ cx+dy=f}…\)

\(\bullet\text{ Identity Matrix}\)
\(\,\,\,\,\,\,\,\,\)\(\left[{\begin{array}{ccc} 1 & 0 & 0\\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} } \right]\)

\(\bullet\text{ Identity and Inverse Matrices}\)
\(\,\,\,\,\,\,\,\,A^{-1}=\displaystyle\frac{1}{ad-bc}\left[{\begin{array}{cc} a & b \\ c & d \\ \end{array} } \right]\)

\(\bullet\text{ Transpose Matrix}\)
\(\,\,\,\,\,\,\,\,\left[{\begin{array}{ccc} 1 \\ 2 \\ 5 \\ \end{array} } \right]\Rightarrow\left[{\begin{array}{c} 1 & 2 & 5 \end{array} } \right]\)

\(\bullet\text{ Rotation Matrix}\)
\(\,\,\,\,\,\,\,\,\)\(R(\theta)=\left[{\begin{array}{cc}\cos{\theta} & -\sin{\theta} \\\sin{\theta} & \cos{\theta} \\\end{array} } \right]\)

\(\bullet\text{ Eigenvectors and Eigenvalues}\)
\(\,\,\,\,\,\,\,\,(A-\lambda I)\vec{v}=\vec{0}\)