Exercise 2.6.17. Find a matrix corresponding to the linear transformation of cyclically permuting vectors in such that applied to produces . Determine the effect of and and explain why .
Answer: We can construct by considering its effect on the elementary vectors in
Applying the given cyclic permutation is equivalent to shifting each entry of a vector up or to the left (depending on how you look at it), so applying to each of through will produce the following vectors respectively:
In other words we must have , , , and .
As discussed in the previous exercise, we can do this by having each column of be the vector into which the corresponding elementary vector should be transformed. In other words, the first column of should be (since transforms into ), the second column should be (since transforms into ), and similarly the third and fourth columns should be and respectively.
We thus have
Consider . We have
Constructing the columns of as we did for we see that
and that has the effect of shifting the entries of a vector up (or to the left) two places:
For we have
Constructing the columns of as we did for and we see that
and that has the effect of shifting the entries of a vector up (or to the left) three places:
Finally for we have
In other words, has the effect of shifting the entries of vectors in up (or to the left) four places, restoring the original vectors.
We then have and . Since is both a left and right inverse of we have .
NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.
If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition, Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books.