## 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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