Eigenvectors

Recall that an eigenvector of a square matrix $A$ is a vector $\vec v$ such that $A\vec v = \lambda \vec v$.

In the figure below, the locations of the four sliders determine the $2\times2$ matrix $A$. The vector $\vec v$ is shown in red; it may be moved by clicking in the head of the vector and dragging it to a new location. The vector $A\vec v$ is shown in gray; it will move in response to a change in $A$ or $\vec v$.

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For instance, set the matrix to $$ A = \left[\begin{array}{cc} 1 & 2 \\ 2 & 1 \end{array}\right]. $$ By interpreting the eigenvector condition geometrically, move the vector $\vec v$ until it is an eigenvector. What is the corresponding eigenvalue?