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

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 .

This entry was posted in linear algebra and tagged , . Bookmark the permalink.

### 1 Response to Linear Algebra and Its Applications, Exercise 3.3.16

1. 赵子萱 says:

You are amazing!!!