## Linear Algebra and Its Applications, Exercise 3.4.5

Exercise 3.4.5. Given a unit vector $u$ and $Q = I - 2uu^T$, prove that $Q$ is orthogonal. What is $Q$ when $u = (\frac{1}{2},\frac{1}{2}, -\frac{1}{2}, -\frac{1}{2})$? $Q^TQ = (I - 2uu^T)^T(I - 2uu^T) = (I^T - 2(uu^T)^T)(I - 2uu^T)$ $= (I - 2(u^T)^Tu^T)(I - 2uu^T) = (I - 2uu^T)(I - 2uu^T)$ $= I \cdot I - 2Iuu^T - 2uu^TI + 4(uu^T)(uu^T) = I - 4uu^T +4u(u^Tu)u^T$ $= I - 4uu^T + 4uIu^T = I - 4uu^T + 4uu^T = I$

Since $Q^TQ = I$ the matrix $Q = I - 2uu^T$ is orthogonal.

If $u = (\frac{1}{2},\frac{1}{2}, -\frac{1}{2}, -\frac{1}{2})$ then $uu^T = \begin{bmatrix} \frac{1}{2} \\ \frac{1}{2} \\ -\frac{1}{2} \\ -\frac{1}{2} \end{bmatrix} \begin{bmatrix} \frac{1}{2}&\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2} \end{bmatrix} = \begin{bmatrix} \frac{1}{4}&\frac{1}{4}&-\frac{1}{4}&-\frac{1}{4} \\ \frac{1}{4}&\frac{1}{4}&-\frac{1}{4}&-\frac{1}{4} \\ -\frac{1}{4}&-\frac{1}{4}&\frac{1}{4}&\frac{1}{4} \\ -\frac{1}{4}&-\frac{1}{4}&\frac{1}{4}&\frac{1}{4} \end{bmatrix}$

so that $Q = I - 2uu^T = \begin{bmatrix} 1&0&0&0 \\ 0&1&0&0 \\ 0&0&1&0 \\ 0&0&0&1 \end{bmatrix} - 2 \cdot \begin{bmatrix} \frac{1}{4}&\frac{1}{4}&-\frac{1}{4}&-\frac{1}{4} \\ \frac{1}{4}&\frac{1}{4}&-\frac{1}{4}&-\frac{1}{4} \\ -\frac{1}{4}&-\frac{1}{4}&\frac{1}{4}&\frac{1}{4} \\ -\frac{1}{4}&-\frac{1}{4}&\frac{1}{4}&\frac{1}{4} \end{bmatrix}$ $= \begin{bmatrix} 1&0&0&0 \\ 0&1&0&0 \\ 0&0&1&0 \\ 0&0&0&1 \end{bmatrix} - \begin{bmatrix} \frac{1}{2}&\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2} \\ \frac{1}{2}&\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2} \\ -\frac{1}{2}&-\frac{1}{2}&\frac{1}{2}&\frac{1}{2} \\ -\frac{1}{2}&-\frac{1}{2}&\frac{1}{2}&\frac{1}{2} \end{bmatrix} = \begin{bmatrix} \frac{1}{2}&-\frac{1}{2}&\frac{1}{2}&\frac{1}{2} \\ -\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&\frac{1}{2} \\ \frac{1}{2}&\frac{1}{2}&\frac{1}{2}&-\frac{1}{2} \\ \frac{1}{2}&\frac{1}{2}&-\frac{1}{2}&\frac{1}{2} \end{bmatrix}$

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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