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## Lecture: Finding Eigenvalues and Eigenvectors (?? minutes)

Give a mini-lecture on how to calculate eigenvalues and eigenvectors. It is often easiest to do this with an example. We like to use the matrix $$A_7\doteq\pmatrix{1&2\cr 9&4\cr}$$ from the Linear Transformations activity since the students have already seen this matrix and know what it's eigenvectors are.

- We like to start by asking the students what they remember from their math classes about eigenvalues and eigenvectors. Typically our students remember that “It has something to do with a determinant and a $\lambda$,” but not much else.
- Introduce the eigenvalue equation. If you have used the Linear Transformations activity, then students should know geometrically that the eigenvectors of a transformation are the vectors that are only changed by a scale. If you get the students to say this in words, you can write the eigenvalue equation $A\vert v\rangle =\lambda \vert v\rangle$ on the board immediately afterwards. This will help them see how the algebraic statement is connected to the geometric statement.
- Demonstrate methods to solve the characteristic equation using your chosen example.
- Demonstrate methods to find the eigenvectors associated with a given eigenvalue using your chosen example.