Multiplying a vector by a matrix is a much more complicated process than multiplying a

Chapter 12, Problem 12.5.15

(choose chapter or problem)

Multiplying a vector by a matrix is a much more complicated process than multiplying a vector by a scalar. However, in certain special cases, matrix and scalar multiplication can lead to the same result. If Av = v where A is a matrix, v a nonzero vector, and a scalar, then v is said to be an eigenvector of A, and is said to be an eigenvalue of A. For instance, let A = 2 1 8 7 , v1 = (1, 8), and 1 = 6. We have: A v1 = 2 1 8 7 1 8 = 6 48 1 v1 = 6 1 8 = 6 48 . We see that multiplying v1 by 1 = 6 works out the same as multiplying v1 by A; we say that v1 is an eigenvector of A with an eigenvalue of 1 = 6. (a) Show that v2 = (1, 1) is an eigenvector of A. What is the eigenvalue? (b) Show that v3 = (3, 3) is an eigenvector of A. What is the eigenvalue? (c) If v is an eigenvector of A, explain why the vectors v and Av are parallel. 16. In this problem, we