Linear Algebra and Its Applications, Exercise 3.3.16

Exercise 3.3.16. Suppose $u$ is a vector with unit length. Show that the matrix $uu^T$ (with rank 1) is a projection matrix.

Answer: We have

$(uu^T)^2 = (uu^T)(uu^T) = u(u^Tu)u^T$

But since $u$ has unit length we have $u^Tu = \|u\|^2 = 1$ so that

$(uu^T)^2 = u \cdot 1 \cdot u^T = uu^T$

We also have

$(uu^T)^T = (u^T)^Tu^T = uu^T$

Since $(uu^T)^2 = uu^T$ and $(uu^T)^T = uu^T$ the rank-1 matrix $uu^T$ is a projection matrix.

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.