Fundamentals of Eigenvalues and Eigenvectors: In-Class Practice

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Section 8.2 Fundamentals of Eigenvalues and Eigenvectors: In-Class Practice

Worksheet 8.2.1 Worksheet
A4 US

In this worksheet we will consider the eigenproblem \(\A\vec{x}=\lambda\vec{x}\text{.}\)

Let \(\A=\left(\begin{array}{rr}18\amp -12\\20\amp -13 \end{array} \right)\text{.}\)
1. Find the characteristic equation of \(\A\)
2. Find the eigenvalues of \(\A.\)
3. For each eigenvalue \(\lambda_i\) in part Item 1.ii, find the eigenspace \(S_{\lambda_i}\)
Repeat the steps in Problem 1 to solve the eigenproblem for \(B=\left(\begin{array}{rrr}5\amp 0\amp 6\\-15\amp -7\amp -3\\-15\amp -12\amp 8\end{array}\right).\)
Let \(\A=\left(\begin{array}{cr} 78 \amp -120\\ 50 \amp -77 \end{array} \right)\text{.}\) \(\A\) has two eigenvalues \(\lambda_1\) and \(\lambda_2\)
1. Find the eigenvalues, \(\lambda_1\) and \(\lambda_2\text{,}\) for \(\A\) and and a corresponding eigenvector for each, \(\vec{v_1}\text{,}\) and \(\vec{v_2}\)
2. Check that \(\lambda_1\) and \(\lambda_2\) are also eigenvalues for \(\A^T\text{.}\)
3. Are the eigenvectors, \(\vec{v_1}\text{,}\) and \(\vec{v_2}\text{,}\) you found in part (a) also eigenvectors for \(\A^T\text{?}\) If not, find eigenvectors for \(\A^T\) and check to see if you can find a relationship.
Let \(\A=\left(\begin{array}{crr} 2\amp -12\amp 36\\0\amp -4\amp 9\\0\amp 0\amp -1 \end{array} \right)\text{.}\)
1. Determine if \(\A\) is diagonalizable. If so, write it as \(\X\Lam\X^{-1}\text{.}\)
2. Use the ideas in this section to find det\((\A)\) using its eigenvalues.
Use the ideas in this section to find a matrix \(\A\) with eigenvalues \(-2, 3\) and corresponding eigenvectors \(\left(\begin{array}{r} -5\\2 \end{array} \right)\) and \(\left(\begin{array}{r}8\\-3 \end{array} \right)\text{,}\) respectively.
Find a matrix \(\A\in \R^{2\times2}\) whose characteristic equation has roots \(5,-3\text{.}\)
Find a matrix \(\A\in \R^{2 \times 2}\) whose eigenvalues are \(1\text{,}\) \(-4\) with corresponding eigenvectors \(\left(\begin{array}{r} 1\\2 \end{array} \right),\left(\begin{array}{r} 0\\-1 \end{array} \right)\text{,}\) respectively.