Exercise 3.3.15. Suppose  is a projection matrix that projects vectors onto a line in the – plane. Describe the effect of the reflection matrix geometrically. Why does ? (Give both a geometric and algebraic explanation.)

Answer: When applied to a vector the matrix produces . This can be thought of combining the following operations:

First, project onto the line in the – plane (). Next, scale the resulting vector by a factor of 2 and reverse its direction (). The resulting vector is still on the line onto which projects. Finally, find the difference between the resulting vector and the original vector (). The final vector and the original vector are symmetric with respect to a line that is perpendicular to the original line onto which projects.

For a simple example, in consider the projection matrix

that projects all vectors onto the axis.

We then have

When is applied to the vector this produces the vector :

In other words, is reflected about the axis, which is perpendicular to the axis onto which projects.

If is applied again then it would reflect the vector back across the axis to produce the original vector . This is true in general for any :

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, Fifth Edition and the accompanying free online course, and Dr Strang’s other books.