Reflector

A reflector (or an elementary reflector) is a special type of unitary matrix that has theoretical and practical utility. A reflector is constructed by taking a vector $$\mathbf u$$ such that $$||\mathbf u||_2 = 1$$ and computing the matrix

$$\mathbf{R = I }-2\mathbf{uu^*}$$

This matrix is called reflector, because the operation $$\mathbf{Rx}$$ reflects any vector $$\mathbf x$$ around the plane $$\mathbf u^\perp$$

Some of the properties of a reflector include:
 * $$\matbhf R^*=\mathbf R^{-1}=\mathbf R$$
 * The property above implies $$\mathbf R^2 = I$$

Reflectors are very useful because they have the unique ability to annihilate all entries except the first one of a given vector $$\mathbf x$$ (see 5.6.10 of C. D. Meyer on how to construct this reflector -- it's simple). This property directly leads to the following:
 * (A practical computation of) the QR decomposition -- see Householder transformation
 * Theoretical proof of existence of SVD (p411 of C.D.Meyer)
 * Showing that eigenvalue/eigenvector computation problem can be simplified through deflation (Example 7.2.6 of C.D.Meyer)
 * Schur's triangulation theorem (p408 of C.D.Meyer)