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{\Large \textbf{STA 302f15 Assignment Three}}\footnote{Copyright information is at the end of the last page.}
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\noindent
For this assignment, Chapter 2 in the text contains material on matrix algebra. You are responsible for what is in this assignment, not everything that's in the text. Questions~\ref{r1} and~\ref{r2} are to be done with R. Please print the two sets of R output on separate pieces of paper. You may be asked to hand one of them in, but not the other.  Except for the R parts, these problems are preparation for the quiz in tutorial on Friday October 1st, and are not to be handed in. 

\emph{Remember that the R parts are \textbf{not group projects}. Do the work yourself. Don't help anybody until you are finished. Don't help anybody who has not started yet. \underline{Never} look at anyone else's code or let anyone look at yours.}

\begin{enumerate} 

    \item In the textbook, do Problems 2.27, 2.28, 2.35, 2.36, 2.37, 2,38 (Prove the ``if" part too), 2.53, 2.76.

    \item \label{r1} In the textbook, do 2.14 a, e, g, and m using R. Show both input (creation of the matrices) and output. Label the output (give letters) using comments. Bring the printout to the quiz.

    \item \label{r2} Make up a your own $4 \times 4$ symmetric matrix that is not singular (that is, the inverse exists), and is \emph{not a diagonal matrix}. If your first try is singular, try again. Call it $\mathbf{A}$. Enter it into R using \texttt{rbind} (see lecture slides). Make sure to display the input. Then, 
        \begin{enumerate}
            \item Calculate $|\mathbf{A}^{-1}|$ and $1/|\mathbf{A}|$, verifying that they are equal.
            \item Calculate $|\mathbf{A}^2|$ and $|\mathbf{A}|^2$, verifying that they are equal.
            \item Calculate the eigenvalues and eigenvectors of $\mathbf{A}$.
            \item Calculate $\mathbf{A}^{1/2}$.
            \item  Calculate $\mathbf{A}^{-1/2}$.
        \end{enumerate} 
Display both input and output for each part. Label the output with comments. Bring the printout to the quiz.  

    \item Recall the definition of linear independence. The columns of 
$\mathbf{X}$ are said to be \emph{linearly dependent} if there exists a $p \times 1$ vector $\mathbf{v} \neq \mathbf{0}$ with $\mathbf{Xv} = \mathbf{0}$. We will say that the columns of $\mathbf{X}$ are linearly \emph{independent} if $\mathbf{Xv} = \mathbf{0}$ implies $\mathbf{v} = \mathbf{0}$. Let $\mathbf{A}$ be a square matrix. Show that if the columns of $\mathbf{A}$ are linearly dependent, $\mathbf{A}^{-1}$ cannot exist. Hint: $\mathbf{v}$ cannot be both zero and not zero at the same time.
    

    \item  \label{ss} Let $\mathbf{a}$ be an $n \times 1$ matrix of real constants. How do you know $\mathbf{a}^\prime\mathbf{a}\geq 0$?

    \item  Recall the \emph{spectral decomposition} of a square symmetric matrix (For example, a variance-covariance matrix).  Any such matrix $\boldsymbol{\Sigma}$ can be written as 
            $\boldsymbol{\Sigma} = \mathbf{CD} \mathbf{C}^\prime$,
where $\mathbf{C}$ is a matrix whose columns are the (orthonormal) eigenvectors of $\boldsymbol{\Sigma}$, $\mathbf{D}$ is a diagonal matrix of the corresponding  eigenvalues, and $\mathbf{C}^\prime\mathbf{C} =~\mathbf{C}\mathbf{C}^\prime =~\mathbf{I}$. 

        \begin{enumerate}
           \item Let $\boldsymbol{\Sigma}$ be a square symmetric matrix with eigenvalues that are all strictly positive. 
                \begin{enumerate}
                    \item What is $\mathbf{D}^{-1}$?
                    \item Show 
            $\boldsymbol{\Sigma}^{-1} = \mathbf{C} \mathbf{D}^{-1} \mathbf{C}^\prime$
                \end{enumerate}

\pagebreak

           \item Let $\boldsymbol{\Sigma}$ be a square symmetric matrix, and this time some of the eigenvalues might be zero. 
                \begin{enumerate}
                    \item What do you think $\mathbf{D}^{1/2}$ might be?
                    \item Define $\boldsymbol{\Sigma}^{1/2}$ as 
                    $\mathbf{CD}^{1/2} \mathbf{C}^\prime$.
                    Show $\boldsymbol{\Sigma}^{1/2}$ is symmetric.
                    \item Show $\boldsymbol{\Sigma}^{1/2}\boldsymbol{\Sigma}^{1/2} = \boldsymbol{\Sigma}$.
                \end{enumerate}

            \item Now return to the situation where the eigenvalues of the square symmetric matrix $\boldsymbol{\Sigma}$ are all strictly positive. Define $\boldsymbol{\Sigma}^{-1/2}$ as $\mathbf{CD}^{-1/2} \mathbf{C}^\prime$, where the elements of the diagonal matrix $\mathbf{D}^{-1/2}$ are the reciprocals of the corresponding elements of $\mathbf{D}^{1/2}$. 
                \begin{enumerate}
                    \item Show that the inverse of $\boldsymbol{\Sigma}^{1/2}$ is $\boldsymbol{\Sigma}^{-1/2}$, justifying the notation.
                    \item Show $\boldsymbol{\Sigma}^{-1/2} \boldsymbol{\Sigma}^{-1/2} = \boldsymbol{\Sigma}^{-1}$.
                \end{enumerate}

           \item The (square) matrix $\boldsymbol{\Sigma}$ is said to be \emph{positive definite} if
           $\mathbf{v}^\prime \boldsymbol{\Sigma} \mathbf{v} > 0$ for all vectors $\mathbf{v} \neq \mathbf{0}$. Show that the eigenvalues of a positive definite matrix are all strictly positive. 
           \item Let $\boldsymbol{\Sigma}$ be a symmetric, positive definite matrix. Putting together a couple of results you have proved above, establish that $\boldsymbol{\Sigma}^{-1}$ exists.
        \end{enumerate}

\item Using the Spectral Decomposition Theorem and $tr(\mathbf{AB})=tr(\mathbf{BA})$, prove that the trace is the sum of the eigenvalues for a symmetric matrix.

\item Using the Spectral Decomposition Theorem and $|\mathbf{AB}| = |\mathbf{BA}|$, prove that the determinant of a symmetric matrix is the product of its eigenvalues.


\end{enumerate}

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This assignment was prepared by  \href{http://www.utstat.toronto.edu/~brunner}{Jerry Brunner},
Department of Statistical Sciences, University of Toronto. It is licensed under a 
\href{http://creativecommons.org/licenses/by-sa/3.0/deed.en_US}
     {Creative Commons Attribution - ShareAlike 3.0 Unported License}. Use any part of it as you like and share the result freely. The \LaTeX~source code is available from the course website:
\href{http://www.utstat.toronto.edu/~brunner/oldclass/302f15} {\small\texttt{http://www.utstat.toronto.edu/$^\sim$brunner/oldclass/302f15}}

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