Scalar Multiplies of Eigenvectors are Eigenvectors

As we said in the videos, an eigenvector of \(A\) associated with a matrix \(A\) can be scaled to meet the needs of the problem at hand. For example, for Markov models, having all of the elements sum to 1 means that the elements are probabilities, and thus have a clear interpretation.

In this exercise, we will be working with the first eigenpair in the previous exercise. For matrix \(A\), this eigenpair has an eigenvalue \(\lambda = 7\) and eigenvector:

          [,1]
[1,] 0.2425356
[2,] 0.9701425
[3,] 0.0000000

Show that double and half of the eigenvector used is still an eigenvector for the given eigenvalue.

Exercise

Scalar Multiplies of Eigenvectors are Eigenvectors

Instructions