Linear Algebra and Its Applications, Exercise 3.4.5

Posted on December 18, 2017 by hecker

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})?

Answer: We have

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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