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 xy 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 xy 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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