## Linear Algebra and Its Applications, Exercise 3.2.9

Exercise 3.2.9. Consider the projection matrix $P = aa^T/a^Ta$ that projects onto a line. Show that $P^2 = P$.

Answer: Note that $a^Ta$ is a scalar (the inner product of $a$ with itself) and $aa^T$ is a matrix. We have $P^2 = \left(aa^T/a^Ta\right)^2 = \left(1/a^Ta\right)^2 \left(aa^T\right)^2$ $= \left(1/a^Ta\right)^2 \left(aa^Taa^T\right) = \left(1/a^Ta\right)^2 a\left(a^Ta\right)a^T$ $= \left(1/a^Ta\right)^2 \left(a^Ta\right)aa^T = \left(1/a^Ta\right) aa^T$ $= aa^T/a^Ta = P$

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 .

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