# Let A be a matrix, a real eigenvalue of A, and v the associated eigen-vector.

Let A be an nxn matrix, λ a real eigenvalue, v the associated eigenvector. Show that A^k v= λ^k v.

Use this to show e^At v= e^λt v. Use this to show that the line through the origin consists of two solutions to x'=Ax.

https://brainmass.com/math/linear-algebra/real-eigenvalue-associated-eigen-vector-343970

#### Solution Summary

Parametrize the two halves of this line.

$2.19