## Linear Algebra and Its Applications, Exercise 3.4.18

Exercise 3.4.18. If $P$ is the projection matrix onto the column space of the matrix $A$ and $A = QR$, what is a simple formula for $P$?

Answer: The projection matrix $P$ onto the column space of $A$ can be calculated as $P = A(A^TA)^{-1}A^T$.

Since the columns of $Q$ are linear combinations of the columns of $A$ the column space of $Q$ is the same as the column space of $A$. Thus $P$ is also the projection matrix onto the column space of $Q$, and we can alternatively calculate $P$ as $P = Q(Q^TQ)^{-1}Q^T$.

But since $Q$ has orthonormal columns we have $Q^TQ = I$ so that $P = QI^{-1}Q^T = QQ^T$. Thus $P = QQ^T$ is a simplified formula for calculating the projection matrix onto the column space of $A = QR$.

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.

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