Exercise 2.6.18. Given a vector in find a matrix that produces a corresponding vector in in which all entries are shifted right one place. Find a second matrix that takes a vector in and produces the vector in in which all entries are shifted left one place. What are the product matrices and and what effects do they have?

Answer: We can construct by considering its effect on the elementary vectors in

Applying to each of through respectively will shift each of their entries to the right (or down, depending on your point of view) and produce the following vectors in :

As discussed in the answer to exercise 2.6.16, we can do this by having each column of be the vector into which the corresponding elementary vector should be transformed. We thus have

so that

Now consider . We can construct by considering its effect on the elementary vectors in

Applying to each of through respectively will shift each of their entries to the left (or up) and produce the following vectors in :

We thus have

so that

Constructing the product matrices we have

and

The product $matrix BA$ corresponds to taking a vector in and shifting the entries right to produce a corresponding vector in and then shifting the entries left to produce the original vector in . So preserves all vectors and is thus equal to the identity matrix on .

On the other hand the product $matrix AB$ corresponds to taking a vector in and shifting the entries left to produce a corresponding vector in and then shifting the entries right to produce the vector in . So the net effect of is to change the first entry of a vector in to zero and preserve the second, third, and fourth entries.

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.