WVWC-AEM Orthogonal Transformations

Section 2.3 Orthogonal Transformations

Two fundamental concepts in vector geometry are the magnitude and the inner product. In \(\RR^n\) these topics are related as given by the following definition.

Definition 2.3.1. Inner Product and Magnitude.

Let \(\vb{x}\) and \(\vb{y}\) be vectors in \(\RR^n\text{.}\) The inner product of \(\vb{x}\) and \(\vb{y}\) is the real scalar \(\dotprod{\vb{x},\vb{y}}\) given by

\begin{equation*} \dotprod{\vb{x},\vb{y}} = \vb{y}^{T}\vb{x}. \end{equation*}

The magnitude of \(\vb{x}\) is the nonnegative real number \(\norm{\vb{x}}\) given by

\begin{equation*} \norm{\vb{x}} = \sqrt{\dotprod{\vb{x},\vb{x}}}. \end{equation*}

If you've taken Calculus III, then some of the following properties will be familiar with their analogues for the dot product (see here ²).

Theorem 2.3.2. Properties of the Inner Product.

Let \(\vb{x}, \vb{y}\) and \(\vb{z}\) be vectors in \(\RR^n\) and let \(\alpha\) and \(\beta\) be real scalars. Then the inner product satisfies the following properties:

\(\displaystyle \dotprod{\vb{x},\vb{y}} = \dotprod{\vb{y},\vb{x}}\)
\(\displaystyle \dotprod{\vb{x}, \vb{y}+\vb{z}} = \dotprod{\vb{x},\vb{y}} + \dotprod{\vb{x},\vb{z}}\)
\(\dotprod{\alpha\vb{x},\vb{y}} = \alpha\dotprod{\vb{x},\vb{y}}\) and \(\dotprod{\vb{x},\beta\vb{y}} = \beta\dotprod{\vb{x},\vb{y}}\)
\(\dotprod{\vb{x},\vb{y}} = \norm{\vb{x}}\norm{\vb{y}}\cos\theta\) where \(\theta\) denotes the angle between \(\vb{x}\) and \(\vb{y}\) where \(0\leq\theta\leq \pi\text{.}\)
\(\dotprod{A\vb{x},\vb{y}} = \dotprod{\vb{x},A^{T}\vb{y}}\) for any \(n\times n\) matrix \(A\text{.}\)

The last property in Theorem 2.3.2 is particularly important and can be taken as the definition of \(A^{T}\text{.}\) We also have the following very important inequalities involving inner products and magnitudes.

Theorem 2.3.3. Cauchy-Schwarz Inequality.

Let \(\vb{x},\vb{y}\in\RR^n\text{.}\) Then

\begin{equation*} \abs{\dotprod{\vb{x},\vb{y}}}\leq\norm{\vb{x}}\norm{\vb{y}}. \end{equation*}

Theorem 2.3.4. Triangle Inequality.

Let \(\vb{x},\vb{y}\in\RR^n\text{.}\) Then

\begin{equation*} \norm{\vb{x} + \vb{y}}\leq\norm{\vb{x}} + \norm{\vb{y}}. \end{equation*}

As the magnitude and inner product are both fundamental concepts in vector geometry, any transformation (i.e., any matrix) that preserves both of these quantities are particularly useful to work with. Such matrices are called orthogonal transformations.

Definition 2.3.5. Orthogonal Transformation.

Let \(U\) be a real \(n\times n\) matrix. We say that \(U\) is orthogonal if

\begin{equation*} UU^{T} = U^{T}U = I. \end{equation*}

The set of all orthogonal \(n\times n\) matrices is denoted by \(O(n)\text{.}\)

Geometrically, the action of an orthogonal matrix on vectors preserves angles between vectors as measured by the inner product.

Theorem 2.3.6. Orthogonal Transformations and Inner Products.

Let \(\vb{x},\vb{y}\in\RR^n\) and let \(U\) be an \(n\times n\) orthogonal matrix. Then

\begin{equation*} \dotprod{\vb{x},\vb{y}} = \dotprod{U\vb{x},U\vb{y}}. \end{equation*}

Furthermore, \(\norm{\vb{x}} = \norm{U\vb{x}}\text{.}\)

As orthogonal matrices are invertible, it follows that their columns form a basis. Such as basis has some very useful characteristics. To be precise, let \(U = \smqty[\vb{u}_1 & \ldots & \vb{u}_{n}]\) denote an \(n\times n\) orthogonal matrix. Then the fact that \(U^{T}U = I\) implies that

\begin{equation*} \dotprod{\vb{u}_{i},\vb{u}_{j}} = \begin{cases} 1 &\text{ if }i=j \\ 0 &\text{ if }i\neq j\end{cases}. \end{equation*}

This leads to the following definition.

Definition 2.3.7. Orthonormal Basis.

Let \(\qty{\vb{u}_{i}}_{i=1}^{n}\) denote a collection of vectors in \(\RR^n\text{.}\) This collection is an orthonormal basis (ONB) if

\begin{equation*} \dotprod{\vb{u}_{i},\vb{u}_{j}} = \begin{cases} 1 &\text{ if }i=j \\ 0 &\text{ if }i\neq j\end{cases}. \end{equation*}

Geometrically, an ONB in \(\RR^n\) is a collection of \(n\) orthogonal unit vectors. These can be viewed as a generalization of the typical coordinate axes.

j-oldroyd.github.io/wvwc-calculus/section-the-dot-product.html

Advanced Engineering Mathematics Lecture Notes: West Virginia Wesleyan College

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