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\pagestyle{fancy} \lhead{{\sf Homework 14 $|$ Due December 3 (Friday) }}
\rhead{\thepage} \cfoot{{\sf Math 8530 $|$ Linear Algebra $|$ Fall
    2021 $|$ M.~Macauley}}


\begin{document}

$\;$

%\noindent\emph{Read}: Lax, Chapter 8, pages 101--120. \\

\begin{enumerate}

  %%------------------------------------------------------------------------
  
\item Consider the following matrices:
  \[
  A=\begin{bmatrix}1&3\\2&6\end{bmatrix},\qquad
  A^TA=\begin{bmatrix}5&15\\15&45\end{bmatrix},\qquad
  AA^T=\begin{bmatrix}10&20\\20&40\end{bmatrix}.
  \]
  \begin{enumerate}
  \item Find the eigenvalues $\sigma_1^2$, $\sigma_2^2$ and unit
    eigenvectors $v_1$, $v_2$ of $A^TA$. 
  \item For the $\sigma_i\neq 0$, compute $u_i=Av_i/\sigma_i$ and
    verify that indeed $||u_i||=1$. Find the other $u_i$ by computing
    the other unit eigenvector of $AA^T$.
  \item Construct the left polar decomposition, $A=UP$. 
  \item Construct the singular value decomposition (SVD),
    $A=U\Sigma V^T$.
  \item Write down orthonormal bases for each the ``four fundamental
    subspaces'': the row space $R_A$, the nullspace $N_A$, the column
    space $R_{A^T}$, and the left nullspace $N_{A^T}$.
  \item Describe \emph{all} matrices that have the same four fundamental
    subspaces.
  \item Find a left inverse, right inverse, and pseudoinverse of $A$,
    or explain why it doesn't exist. \bigskip
\end{enumerate}
  
%%------------------------------------------------------------------------
  
\item Compute the polar and singular value decomposition of the
  rotation matrix
  \[
  A=\begin{bmatrix}a & -b \\ b & a\end{bmatrix}
  \]
  where $a,b\in\R$. \bigskip
  
  %%------------------------------------------------------------------------
  
\item Consider the matrix
  \[
  A=\begin{bmatrix} 1 & 0 & 1 \\ -1 & 1 & 0 \end{bmatrix}.
  \]
  \begin{enumerate}
  \item Construct the singular value decomposition of $A$.
  \item Write down orthonormal bases for each the ``four fundamental
    subspaces'': the row space $R_A$, the nullspace $N_A$, the column
    space $R_{A^T}$, and the left nullspace $N_{A^T}$.
  \item Find a left inverse, right inverse, and pseudoinverse of $A$,
    or explain why it doesn't exist.
  \end{enumerate}
    
  %%------------------------------------------------------------------------

\end{enumerate}

\end{document}







  
