## Linear Algebra and Its Applications, Exercise 3.3.9

Exercise 3.3.9. Suppose that $P$ is a matrix such that $P = P^TP$.

a) Show that $P$ is a projection matrix.

b) If $P = 0$ then what is the subspace onto which $P$ projects?

Answer: a) To show that $P$ is a projection matrix we must show that $P = P^2$ and also that $P = P^T$. We have $P^T = (P^TP)^T = P^T(P^T)^T = P^TP = P$

Since $P^T = P$ we then have $P^2 = P P = P^T P = P$

Since $P = P^T = P^2$ the matrix $P$ is a projection matrix.

b) If $P = 0$ then for all vectors $v$ we have $P v = 0$. So $P$ projects onto the subspace consisting of the zero vector.

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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