## 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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