## Linear Algebra and Its Applications, Exercise 3.2.13

Exercise 3.2.13. For a projection matrix $P = aa^T/a^Ta$ show that the sum of the diagonal entries of $P$ (the “trace” of $P$) always equals one.

Answer: For $a = \left( a_1, a_2, \dots, a_n \right)$ the diagonal entries of $P = aa^T/a^Ta$ are $a_1^2 / (a^Ta)$, $a_2^2 / (a^Ta)$, through $a_n^2 / (a^Ta)$ so that the trace of $P$ is $\left( a_1^2 + a_2^2 + \cdots + a_n^2 \right) / (a^Ta) = a^Ta / (a^Ta) = 1$

UPDATE: Corrected the formulas for the diagonal entries of $P$.

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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