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\begin{center}
  {\bf University of Kentucky, Physics 306 \\
    Homework \#4, Rev. B, due Monday, 2023-02-13 }
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\vspace{.3 in}


{\bf 1. Similarity transforms and invariants} of the Pauli matrices $\sigma_{x}, \sigma_{y}, \sigma_{z}$ and in general.

{\bf \quad a)} What are the eigenvectors and eigenvalues of $\sigma_{z}$?  Diagonalize $\sigma_{x}$ and $\sigma_{y}$ to obtain their eigenbasis $U_{x}=\frac{1}{\sqrt2}\pmatrix{1 & 1\cr 1 &\!\!\!\! -1}$ and $U_{y}=\frac{1}{\sqrt2}\pmatrix{1 & 1\cr i &\!\!\!\! -i}$.  Verify that $U_x$ and $U_{y}$ are unitary. % $U_{x}\ad U_{x}=U_{y}\ad U_{y}=I$, i.e. their columns (the eigenvectors of $\sigma_{x}$ and $\sigma_{y}$) are orthonormal.

{\bf \quad b)} Justify the eigenequations $\sigma_{x}U_{x}=U_{x}\sigma_{z}$ and $\sigma_{y}U_{y}=U_{y}\sigma_{z}$.  For $i,j{=}x,y,z$, find the matrices $U_{ij}$ of the {\it similarity transforms} $\sigma_{i}=U_{ij}\sigma_{j}U_{ij}\ad$ (the ${}_{i}$, ${}_{j}$,  and ${}_{ij}$ are not component indices!).

{\bf \quad c)} [bonus: The {\it trace} of a matrix is the sum of its diagonal elements: $\mathrm{tr} A=A_{ii}$.  Calculate $\mathrm{tr}(0)$ and $\mathrm{tr}(\sigma_i)$.  Show that $\mathrm{tr}(ABC)=\mathrm{tr}(BCA)=\mathrm{tr}(CAB)$ but not necessarily $\mathrm{tr}(ABC)=\mathrm{tr}(CBA)$.  Thus if $A'=UAU\ad$ then $\mathrm{tr}(A')=\mathrm{tr}(A)$, i.e. the trace, or `perimeter', of $A$ is invariant.]

{\bf \quad d)} [bonus: The {\it determinant} of a matrix is the fully antisymmetric product of one element in each row or column: $\mathrm{det} A=\varepsilon_{ijk}A_{1i}A_{2j}A_{3k}=\varepsilon_{ijk}A_{i1}A_{j2}A_{k3}$.  Note that $\mathrm{det}(A\ad)=\mathrm{det}(A)^{*}$ and $\mathrm{det}(AB)=\mathrm{det}(A)\mathrm{det}(B)$.  Calculate $\mathrm{det}(I)$ and $\mathrm{det}(\sigma_i)$.  Show that if $U\ad U=I$ then $|\mathrm{det}(U)|=1$, and if $A'=UAU\ad$, then $\mathrm{det}(A')=\mathrm{det}(A)$, i.e. the determinant, or `volume', of $A$ is invariant.  Show that $\exp(\mathrm{tr}(A))=\mathrm{det}(\exp(A))$, for example, $\exp(\mathrm{tr}(0))=1=\mathrm{det}(\exp(0)=I)$.]

{\bf \quad e)} [bonus: Show that the {\it characteristic equation} $|A-\lambda I| = \lambda^{n}+a_{n-1}\lambda^{n-1}+\ldots+a_{1}\lambda+a_{0}=0$, substituting $\lambda\to A$ in the second equality, is invariant under similarity transforms.  Thus $A$ has $n$ independent invariants.  The first and last coefficients $a_{n-1}$ and $a_{0}$ are $\mathrm{tr}(A)$  and $\mathrm{det}(A)$, respectively, while the others are $k$-dimensional `perimeters', for example, {\it surface area}, of the transformation.]

{\bf \quad f)} [bonus: Prove the \href{https://www.cuemath.com/algebra/cayley-hamilton-theorem/}{Cayley-Hamilton Theorem} (that any matrix $A$ satisfies its own characteristic equation) for  the case of diagonal matrices.  The full theorem follows from the invariance of the characteristic equation.  Show that $A^{-1}=-(A^{n-1}+a_{n-1}A^{n-2}+\ldots+a_{1}I)/a_{0}$.]


{\bf 2.  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.  This problem explores the structure of stretches.

{\bf \quad a)} 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 $S \bs{v}_{i}= \bs{v}_{i}\lambda_{i}$.  Augment these $n$ equations to obtain $SV=VW$, where $W=\rm{diag}(\lambda_{1},\lambda_{2},\ldots)$, and thus show $S=VWV\ad$ and $W=V\ad SV$.

{\bf \quad b)} Show that the eigenvalues of $S$ must be real: $\lambda^{*}=\lambda$, and 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 thus show the matrix of eigenvectors is unitary: $V\ad V=I$.  [bonus: Interpret the similarity transforms geometrically.]

{\bf \quad c)} 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 H03\#1b).  This is an example of the \href{https://en.wikipedia.org/wiki/Normal_matrix#Normal_matrix_analogy}{\it normal matrix analogy}.
% , which connects matrices and complex numbers.
%Hermitian matrices have real eigenvalues while anti-Hermitian matrices have zero trace and imaginary eigenvalues.  The exponential of a Hermitian matrix is positive definite and has real positive eigenvalues, while the exponential of an anti-Hermitian matrix is unitary with  unit modulus determinant and eigenvalues.  


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