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
This exercise is part of the course
Linear Algebra for Data Science in R
Exercise instructions
- Show that double and half of the eigenvector used is still an eigenvector for the given eigenvalue.
Hands-on interactive exercise
Have a go at this exercise by completing this sample code.
# Show that double an eigenvector is still an eigenvector
A%*%((___)*c(0.2425356, 0.9701425, 0)) - 7*(___)*c(0.2425356, 0.9701425, 0)
# Show half of an eigenvector is still an eigenvector
___%*%((0.5)*c(0.2425356, 0.9701425, 0)) - ____*(0.5)*c(0.2425356, 0.9701425, 0)