19.2: Solving Eigenproblems - A 2x2 Example

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            'answercheck.py');

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Consider calculating eigenvalues for any \(2 \times 2\) matrix. We want to solve:

\[|A - \lambda I_2 | = 0 \nonumber \]

\[\begin{split}
\left|
\left[
\begin{matrix}
a_{11} & a_{12} \\
a_{21} & a_{22}
\end{matrix}
\right]
- \lambda \left[
\begin{matrix}
1 & 0 \\
0 & 1
\end{matrix}
\right]
\right|
=
\left|
\left[
\begin{matrix}
a_{11}-\lambda & a_{12} \\
a_{21} & a_{22}-\lambda
\end{matrix}
\right]
\right|
=0
\end{split} \nonumber \]

We know this determinant:

\[(a_{11}-\lambda)(a_{22}-\lambda) - a_{12} a_{21} = 0 \nonumber \]

If we expand the above, we get:

\[a_{11}a_{22}+\lambda^2-a_{11}\lambda-a_{22}\lambda - a_{12} a_{21} = 0 \nonumber \]

and

\[\lambda^2-(a_{11}+a_{22})\lambda+a_{11}a_{22} - a_{12} a_{21} = 0 \nonumber \]

This is a simple quadratic equation. The roots of \(A\lambda^2 + B\lambda + C = 0 \) can be solved using the quadratic formula:

\[ \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \nonumber \]

Question

Using the above equation. What are the eigenvalues for the following \(2 \times 2\) matrix. Try calculating this by hand and then store the lower value in a variable namede1 and the larger value in e2 to check your answer:

\[\begin{split}A =
\left[
\begin{matrix}
-4 & -6 \\
3 & 5
\end{matrix}
\right]
\end{split} \nonumber \]

# Put your answer here

from answercheck import checkanswer

checkanswer.float(e1,'c54490d3480079138c8c027a87a366e3');

from answercheck import checkanswer

checkanswer.float(e2,'d1bd83a33f1a841ab7fda32449746cc4');

Do This

Find a numpy function that will calculate eigenvalues and verify the answers from above.

# Put your answer here

Question

What are the corresponding eigenvectors to the matrix \(A\)? This time you can try calculating by hand or just used the function you found in the previous answer. Store the eigenvector associated with the e1 value in a vector named v1 and the eigenvector associated with the eigenvalue e2 in a vector named v2 to check your answer.

# Put your answer here

from answercheck import checkanswer

checkanswer.eq_vector(v1,'35758bc2fa8ff4f04cfbcd019844f93d');

from answercheck import checkanswer

checkanswer.eq_vector(v2,'90b0437e86d2cf70050d1d6081d942f4');

Question

Both sympy and numpy can calculate many of the same things. What is the fundamental difference between these two libraries?