% begin preamble
\documentclass[11pt,titlepage]{article}
\usepackage[margin=1in,noheadfoot]{geometry}
\usepackage[parfill]{parskip}
\usepackage{graphicx}
\usepackage[colorlinks,urlcolor=blue]{hyperref}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{wrapfig}

\newcommand{\e}[1]{\times 10^{#1}}
\newcommand{\q}{\mathrm}
\newcommand{\where}[1][where]{\qquad\mbox{#1}\qquad}
\newcommand{\bs}[1]{\boldsymbol{#1}}
\newcommand{\bh}[1]{\hat{\boldsymbol{#1}}}
\newcommand{\bv}[1]{\vec{\boldsymbol{#1}}}
\newcommand{\bt}[1]{\tilde{\boldsymbol{#1}}}
\newcommand{\ad}{^{\dagger}}
\newcommand{\flow}[1]{\mathcal{E}_{#1}}
\newcommand{\flux}[1]{\Phi_{#1}}
\newcommand{\dslash}{d \hspace{-0.5ex}\rule[1.2ex]{0.6ex}{.1ex}}
\newcommand{\ts}{\textsuperscript}
\def\rcurs{{\mbox{$\resizebox{.16in}{.08in}{\includegraphics{ScriptR}}$}\!\!}}
\def\brcurs{{\mbox{$\resizebox{.16in}{.08in}{\includegraphics{BoldR}}$}\!\!}}
\def\hrcurs{{\mbox{$\hat \brcurs$}\!}}
%end preamble

\begin{document}
\begin{center}
  {\bf University of Kentucky, Physics 404G \\
    Homework \#2, Rev. A, due Tuesday, 2021-09-14 }
\end{center}

\vspace{.3 in}

{\bf 1. Projections}: relational versus parametric geometry.

{\bf \quad a)} Show graphically that the following equations define a set of points $\{\bs x\}$ \mbox{on a {\bf line} or {\bf plane},}
\begin{center}
  \begin{tabular}{r|@{\qquad}l@{\qquad}l}
    & relational & parametric \\ \hline
    line & $\{\bs x ~|~ \bs a \times \bs x = \bs d\}$  &  $\{\bs x = \bs x_1 + \bs a \alpha \phantom{+ \bs c \gamma} ~~|~ \quad \alpha \in \mathbb{R}\}$ \\
    plane& $\{\bs x ~|~ \bs A \cdot \bs x = D\}$  &  $\{\bs x = \bs x_2 + \bs b \beta + \bs c \gamma ~|~ \beta,\gamma \in \mathbb{R}\}$ \\
  \end{tabular}
\end{center}
where $\bs a, \bs d, \bs A, D$ are constants which define the geometry, $\bs x_{1,2}$ are fixed points on the line and plane respectively, and $\alpha, \beta, \gamma$ are parameters that vary along the line/plane (they uniquely parametrize points in the line/plane).  [bonus:] Show the 5th relation $\{\bs x = \bs x_2 + \bs A \times \bs \delta ~|~ \bs \delta \in \mathbb{R}^{3}\}$.

{\bf \quad b)} What constraint between $\bs a$ and $\bs d$ is implicit in the formula $\bs a \times \bs x = \bs d$?  Likewise, what relations between $\bs b$, $\bs c$, and $\bs A$ must be satisfied if both equations describe the same plane?  For the line and the plane, substitute $\bs x$ from the parametric form into the relational form to show that they are consistent.  What are $\bs d$ and $D$ in terms of $\bs a, \bs x_1$ and $\bs A, \bs x_2$, respectively?

{\bf \quad c)} We define $\bt a \equiv \bs A /( \bs a{\cdot} \bs A)$.  It is parallel to $\bs A$ but {\it normalized} in the sense that $\bs a \cdot \bt a = 1$.  Using the BAC-CAB rule, show that $\bs x = \bs a (\bt a\cdot \bs x) - \bt a \times (\bs a \times \bs x)$ for any $\bs x$.  This is a non-orthogonal projection of $\bs x$ into a vector parallel to the line and a vector parallel to the plane.  Illustrate this projection and show which term corresponds to each.

{\bf \quad d)} [bonus:] Using c), calculate the point $\bs x_0$ at the intersection of the line and plane in terms of $\bs a, \bs d, \bs A, D$.  Verify this by showing $\bs x_0$ satisfies the relational \mbox{equation for both the line and plane.}

{\bf \quad e)} Let ~$\bt a = \frac{\bv b\times \bv c}{\bv a \cdot \bv b \times \bv c}$, ~$\bt b = \frac{\bv c\times \bv a}{\bv a \cdot \bv b \times \bv c}$,~and ~$\bt c = \frac{\bv a\times \bv b}{\bv a \cdot \bv b \times \bv c}$, where the arrows on $\bs a, \bs b, \bs c$ distinguish $\bv a, \bv b, \bv c$ from their {\it covectors} $\bt a, \bt b, \bt c$.  This definition of $\bt a$ is consistent with above if $\bs A = \bv b \times \bv c$.  Calculate the nine combinations of $(\bv a, \bv b, \bv c)^T \cdot (\bt a, \bt b, \bt c)=I$, i.e.  $\bv a \cdot\bt a=1$, $\bv a \cdot\bt b=0$, \ldots to show they are {\it mutually orthogonal}.  In this sense, $(\bt a, \bt b, \bt c)$ is the \href{https://en.wikipedia.org/wiki/Dual_basis}{\it dual} or \href{https://en.wikipedia.org/wiki/Dual_basis}{\it reciprocal} basis of $(\bv a, \bv b, \bv c)$.

{\bf \quad f)} The {\it contravariant components} of $\bs x$ are defined as the components $(\alpha,\beta,\gamma)$ that satisfy the equation $\bs x = \bv a \alpha + \bv b \beta + \bv c \gamma$; i.e., $(\bv a, \bv b, \bv c)$ is the contravariant basis.  Using $\bs x \cdot \bt a$, etc., calculate the three contravariant components of $\bs x$ in terms of dot products.  Likewise, find the {\it covariant components} $(\tilde\alpha,\tilde\beta,\tilde\gamma)$ of $\bs x$, defined as the components $(\tilde\alpha,\tilde\beta,\tilde\gamma)$ which satisfy the equation $\bs x = \bt a \tilde\alpha + \bt b \tilde\beta + \bt c \tilde\gamma$; i.e.  $(\bt a, \bt b, \bt c)$ is the covariant basis.


{\bf 2. Rotations}.  In addition to projection, the cross product
\href{https://en.wikipedia.org/wiki/Angular_displacement#Generators_of_rotations}{\it
  generates rotation}.  For example, $\bs\omega\times$ transforms to a
rotating reference frame and $\bs r\times$ appears in torque and angular
momentum.  Rotations are ubiquitous in all of physics, particularly mechanics.
We will explore parallels between the cross product, the unit imaginary $i$,
and matrix generator $M$ in the context of rotations.

{\bf \quad a)} The {\it complex plane} $\{\bs w=(x,y)\}$ is the vector space of {\it real} and {\it imaginary} components of complex numbers $w=x + iy \in \mathbb{C}$, where $i^{2}=-1$.  Identify the two natural {\it basis vectors} (complex numbers) in this space.  In addition to linear structure, show that complex multiplication has the form $w=w_{1}w_{2}=(x_{1}x_{2}-y_{1}y_{2})+i(x_{1}y_{2}+y_{1}x_{2})=x+iy$, matching matrix \mbox{products of \scalebox{.8}{$\pmatrix{x & -y \cr y & x}$}.}

{\bf \quad b)} Show that the dot and cross product of two points $\bs w_1=(x_1,y_1)$ and $\bs w_2=(x_2,y_2)$ are given by the real and imaginary parts of the complex product $w_1^* w_2 = \bs w_1\cdot \bs w_2 + i (\bs w_1 \times \bs w_2)_z$, where $w^*=x-iy$ is the {\it complex conjugate} of $w$.  Identify the symmetric and antisymmetric terms of this product.  Thus the complex product $|w|^2 = w^*w$ equals the vector square $\bs w \cdot \bs w$.

{\bf \quad c)} Show graphically that the operator $w\to iw$ rotates the point $w$ 90$^\circ$ CCW about the origin, and that the operator $1+i\, d\phi: w \mapsto w +i w\, d\phi$ preserves the magnitude of $w$ (assuming $d\phi\,^2=0$), but rotates it CCW by the infinitesimal angle $d\phi$.

{\bf \quad d)} Obtain a finite rotation from an infinite number of $d\phi$ rotations as follows: formally integrate the equation $dw = i w\, d\phi$ with the initial condition $w|_{\phi=0}=w_0$ to obtain the rotation formula $w(\phi)=R_\phi w_0$, where $R_\phi = e^{i\phi}$.  Use this result to justify the identity {$\displaystyle e^x=\lim_{n\to\infty}\textstyle(1+\frac{x}{n})^n$}.  Show this is a rotation by separating the Taylor expansion of $e^{i\phi}$ into $x+iy$ to prove Euler's formula, $e^{i\phi}=\cos\phi + i \sin\phi$.

{\bf \quad e)} Show that complex multiplication by $i$ is equivalent to the vector operator $\bh z\, \times$, and determine the matrix representation $M_z$ of this operator; ie. $M_z \bs r = \bh z \times \bs r$.  Do the same for $M_x$ and $M_y$ to show that $\bs v\times = \bs v \cdot \bs M = v_x M_x + v_y M_y + v_z M_z$ is the matrix representation of $\bs v\, \times$ for any vector $\bs v$.  {\it Answer:} $\bs M \sim (M_i)_{jk}=\varepsilon_{ijk}$, the Levi-Civita (cross product), tensor which is completely antisymmetric in indices $i,j,k$.

{\bf \quad f)} Restricted to the $xy$-plane, show that $M_z^2=-I$, a $2\times 2$ matrix, analogous to $i^2=-1$, and thus the matrix for a CCW rotation $\phi$ is $R_\phi = e^{M_z \phi} = I \cos\phi + M_z \sin\phi =$ \scalebox{.8}{${\pmatrix{\cos\phi & -\sin\phi\cr \sin\phi & \cos\phi}}$}.  Note that the {\it exponential} of a matrix $M_z$ is defined by its Taylor expansion.  The \href{https://en.wikipedia.org/wiki/Rodrigues\%27_rotation_formula}{\it Rodrigues formula} for the matrix $R_{\bs v}$ of a CCW rotation by an angle $v=|\bs v|$ about the $\bh v$-axis is $R_{\bs v} = e^{\bs M \cdot \bs v} = I \cos v + M{\cdot} \bh v \sin v + \bh v \bh v^T (1-\cos v)$. The third term projects out the axis of rotation $\bh v$.


{\bf 3.  Stretches}.  In analogy with the \href{https://en.wikipedia.org/wiki/Polar_decomposition#Matrix_polar_decomposition}{\it polar decomposition} $w=x+iy=\rho e^{i\phi}$ of complex numbers, any matrix $A$ can be decomposed $A=RS$ into a stretch $S$ and a rotation $R$, which are the building blocks of all linear operators.

{\bf \quad a)} Use the exponential form $R_{\bs v} = e^{\bs M \cdot \bs v}$ and the asymmetry of the generator $M\ad=-M$ to show that $R\ad R=I$ or $R\ad=R^{-1}$.  In contrast, a stretch is symmetric or {\it Hermitian} $S\ad=S$.

{\bf \quad b)} A symmetric matrix $S$ is guaranteed to have a complete set of eigenvectors $V=(\bs{v}_{1}, \bs{v}_{2}, \ldots )$ and corresponding eigenvalues $\lambda_{1}, \lambda_{2}, \ldots$, such that $S \bs{v}_{i}= \bs{v}_{i}\lambda_{i}$.  Show that we can extend this equation to $SV=VW$, where $W=\rm{diag}(\lambda_{1},\lambda_{2},\ldots)$ is {\it diagonal}.

{\bf \quad c)} Show that two eigenvectors $\bs{v}_{i},\bs{v}_{j}$ of $S$ with distinct eigenvalues $\lambda_{i}\neq\lambda_{j}$ must be orthogonal: $\bs{v}_{i}\cdot\bs{v}_{j}=0$, and therefore the matrix of eigenvectors is an orthogonal transformation: $V\ad V=I$.  [bonus]: Interpret the resulting decompositions $S=VWV\ad$ and $W=V\ad SV$ \mbox{in terms of rotations.}

{\bf \quad d)} Calculate the eigenvalues and eigenvectors of $M_z=$\scalebox{.8}{$\pmatrix{0&-1\cr 1&0}$} Do they look familiar?  Show that $M_z = VWV\ad = \frac12$\scalebox{.8}{$\pmatrix{1&1 \cr i&-i} \pmatrix{-i&0\cr0&i} \pmatrix{1&-i\cr1&i}$} and $e^{M_z \phi} = V e^{W\phi}V\ad = \frac12$\scalebox{.8}{$\pmatrix{1&1 \cr i&-i} \pmatrix{e^{-i\phi}&0\cr0&e^{i\phi}} \pmatrix{1&-i\cr1&i}$}.  Multiply this out to verify part 2f).  Thus real Hermitian matrices have real eigenvalues while antiHermitian matrices have Tr=0 and imaginary eigenvalues.  The exponential of a Hermitian matrix is positive definite with real positive eigenvalues, while the exponential of an antiHermitian matrix is unitary with Det=1 and unit modulus eigenvalues.  This is the normal matrix analogy which completes the connection between matrices and complex numbers.

{\bf \quad e)} \href{https://en.wikipedia.org/wiki/Singular_value_decomposition}{\it Singular Value Decomposition (SVD)} [bonus] Combine \mbox{polar ($A=RS$) and eigen ($S=VWV\ad$)} decompositions, using $U=RV$, to show that any $m\times n$ matrix $_{m}A_{n}$ may be decomposed into \mbox{$A=UWV\ad$}, where $U\equiv V\ad R$,~ $U\ad U=I_{m}$, $V\ad V=I_{n}$, and $_{m}W_{n}$ is diagonal and \mbox{\it positive definite}.

\end{document}

%%% Local Variables:
%%% mode: latex
%%% TeX-master: t
%%% End:
