1. Subtract Lambda: Use your eigenvalue to find the shifted matrix $(A - Λ I)$.
2. Row Check: The two rows of the shifted matrix must be dependent (one is a multiple of the other).
3. Find the Ratio: Solve the equation $ax + by = 0$.
4. The "Swap" Shortcut: A simple eigenvector is always $\mathbf{v} = \langle b, -a \rangle$.
For each matrix and given eigenvalue, find an associated eigenvector $\mathbf{v}$.
$A = \begin{bmatrix} 2 & 1 \\ 3 & 4 \end{bmatrix}$ with $\lambda = 5$.
$A = \begin{bmatrix} 3 & 4 \\ 4 & 3 \end{bmatrix}$ with $\lambda = 7$.
$A = \begin{bmatrix} 5 & 0 \\ 0 & 2 \end{bmatrix}$ with $\lambda = 5$.
Take your eigenvector $\mathbf{v}$ from Problem 1. Multiply it by the Original Matrix $A$. Show the calculation. Do you get $\langle 5, 15 \rangle$? Why is the result exactly 5 times larger than your vector? What does this tell you about the "Agreement" between the Matrix and the Direction?
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For $A = \begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}$, we found $\lambda = 3$ and $\lambda = -1$.
1. Find the eigenvector for $\lambda = 3$.
2. Find the eigenvector for $\lambda = -1$.
Find the eigenvectors for the Identity Matrix $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ with $\lambda = 1$.
Task: Set up the shift $(I - 1I)$.
Does it matter what vector you pick?
Why is every direction an eigenvector for the Identity?
Objective: Explain Eigenvectors to a younger student using a flashlight and a shadow.
The Activity:
1. Hold a pencil straight up and down. Shine the light from above.
2. Move the light left and right. The shadow moves.
3. "Now, point the pencil directly at the light. No matter how much I shake the light, the pencil stays pointed at the source."
The Lesson: "An Eigenvector is a part of us that stays pointed at God even when everything else is shaking. It's our 'Fixed Focus'."
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