Linear Algebra with Applications, 10th Edition

7.5 Orthogonal Transformations

Orthogonal transformations are one of the most important tools in numerical linear algebra. The types of orthogonal transformations that will be introduced in this section are easy to work with and do not require much storage. Most important, processes that involve orthogonal transformations are inherently stable. For example, let $\mathbf{x}\in {ℝ}^n$ and ${\mathbf{x}}^{\prime }=\mathbf{\text{x+}}\mathbf{e}$ be an approximation to x: If Q is an orthogonal matrix, then

$$Q{\mathbf{x}}^{\prime }=Q\mathbf{\text{x+}}Q\mathbf{e}$$

The error in $Q{\mathbf{x}}^{\prime }$ is Qe. With respect to the 2-norm, the vector Qe is the same size as e;

$${\Vert Q\mathbf{e}\Vert }_2={\Vert \mathbf{e}\Vert }_2$$

Similarly, if $A^{\prime }=A+E$, then

$$Q{A}^{\prime }=QA+QE$$

and

$$\Vert {QE\Vert }_2={\Vert E\Vert }_2$$

When an orthogonal transformation is applied to a vector or matrix, the error will not grow with respect to the 2-norm.

Elementary Orthogonal ...