## Linear Algebra and Its Applications, Exercise 3.3.15

Exercise 3.3.15. Suppose $P$ is a projection matrix that projects vectors onto a line in the $x$ $y$ plane. Describe the effect of the reflection matrix $H = I-2P$ geometrically. Why does $H^2 = I$? (Give both a geometric and algebraic explanation.)

Answer: When applied to a vector $v$ the matrix $H$ produces $Hv = (I-2P)v = v - 2Pv$. This can be thought of combining the following operations:

First, project $v$ onto the line in the $x$ $y$ plane ( $Pv$). Next, scale the resulting vector by a factor of 2 and reverse its direction ( $-2Pv$). The resulting vector is still on the line onto which $P$ projects. Finally, find the difference between the resulting vector and the original vector $v$ ( $v - 2Pv$). The final vector $Hv$ and the original vector $v$ are symmetric with respect to a line that is perpendicular to the original line onto which $P$ projects.

For a simple example, in $\mathbb{R}^2$ consider the projection matrix $P = \begin{bmatrix} 1&0 \\ 0&0 \end{bmatrix}$

that projects all vectors onto the $x$ axis.

We then have $H = I -2P = \begin{bmatrix} 1&0 \\ 0&1 \end{bmatrix} - 2 \cdot \begin{bmatrix} 1&0 \\ 0&0 \end{bmatrix}$ $= \begin{bmatrix} 1&0 \\ 0&1 \end{bmatrix} - \begin{bmatrix} 2&0 \\ 0&0 \end{bmatrix} = \begin{bmatrix} -1&0 \\ 0&1 \end{bmatrix}$

When $H$ is applied to the vector $(2, 3)$ this produces the vector $(-2, 3)$: $\begin{bmatrix} -1&0 \\ 0&1 \end{bmatrix} \begin{bmatrix} 2 \\ 3 \end{bmatrix} = \begin{bmatrix} -2 \\ 3 \end{bmatrix}$

In other words, $(2, 3)$ is reflected about the $y$ axis, which is perpendicular to the $x$ axis onto which $P$ projects.

If $H = I - 2P$ is applied again then it would reflect the vector $(-2, 3)$ back across the $y$ axis to produce the original vector $(2, 3)$. This is true in general for any $H = I-2P$: $H^2 = (I-2P)(I-2P) = I^2 - 2IP - 2PI + 4P^2$ $= I - 2P -2P + 4P = I$

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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